A.STRONOMY  DEPT 


INVESTIGATION  OF  INEQUALITIES  Efc* 

s^*/ 

MOTION  OF  THE  MOON  PRODUCED  BY 
THE  ACTION  OF  THE  PLANETS 


SIMON  NEWCOMB 


ASSISTED    BY 


FRANK  E.  ROSS 


WASHINGTON,  D.  C.: 

PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OF  WASHINGTON 

JUNE,  1907 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE    III.  —Concluded. 
MUTUAL  PERIODIC  PERTURBATIONS  OF  VENUS  AND  THE  EARTH. 

The  term  of  long  period  is  omitted.     The  tabular  unit  is  o".oi  in  tu  and  <5f',  and  10— 8  in  Sp  and  <)//. 


System  9. 

System^  10. 

System  ii. 

i 

du 

8v' 

dp 

Sto> 

du 

dv' 

dp 

V 

du 

dv' 

fy 

8p> 

0 

-  508 

+447 

+  196 

+563 

-  548 

+407 

+  300 

+495 

-  205 

+  158 

+  341 

+471 

I  1  —  544 

+3i8 

+  174 

+583 

-  616 

+355 

+  295 

+506 

-  371 

+  169 

+  372 

+454 

2   —  482 

+  176   +  237 

+552 

—  660  j  +285 

+  355 

+469 

—  540 

+  175 

+  468 

+393 

3 

—  444 

+  36   +  357 

+487 

-698 

+200 

+  465 

+398 

-  718 

+  179 

+  591 

+306 

4 

—  446 

—  90 

+  493 

+394 

—  747 

+  109 

+  579 

+301 

-897 

+  176 

+  700 

+207 

5   —  493 

—196   +  613 

+285 

-  811 

+  19 

+  673 

+  193 

-1066 

+  165 

+  767 

+  103 

6   —  576 

-278   +  699 

+162 

-889 

-  62 

+  727 

+  82 

—  1208 

+  141 

+  783 

+  o 

7   -690 

-338  i  +  744 

+  26 

—  980 

-127 

+  732 

—  32 

—  1321 

+  109 

+  757 

-104 

8 

-  828 

—370 

+  736 

—  121 

—1073 

—  171 

+  681 

—142 

—  1409 

+  77 

+  662 

-205 

9 

-981 

—371 

+  667 

—272 

—  1164 

—  194 

+  577 

-254 

-1470 

+  51 

+  5i6 

-300 

10   —1138 

—335 

+  522 

—42O 

—1249 

—  193 

+  4i6 

—364 

-1497 

+  39 

+  342 

-384 

ii    —1274 

—260 

+  298 

-553 

—1290 

-166 

+  203 

-472 

—  1484 

+  41 

+  H4 

-456 

12     —1360 

—  147 

—   3 

—  661 

—1310 

—  113 

—  60 

-569 

-1423 

+  59 

—  147 

—517 

13     —1372 

—  i 

—  351 

—740 

—1274 

-  29 

-  364 

-648 

-1314 

+  95 

—  427 

-566 

M 

-1288 

+  164 

-  715 

-783 

—1176 

+  83 

-693 

-699 

—  1154 

+  147 

—  7H 

-603 

IS 

—  1106 

+339 

—1055 

-786 

—  IOOO 

+222 

—1015 

-716 

-  938 

+220 

-987 

—624 

16 

-835 

+504 

-1328 

-753 

—  741 

+374 

—  1301 

-695 

-  669 

+311   —1237 

—620 

17 

—  493 

+654 

—  1520 

-682 

-  415 

+530 

—  1511 

-637 

—  346 

+420   —1438 

-583 

18 

-  106 

+779 

—1609 

-576 

—  40 

+671 

—  1617 

—549 

+  19 

+536   -1561 

-Sio 

19 

+  308 

+876 

-1582 

—440 

+  353   +791 

—  1605 

-431 

+  413 

+651   —1579 

—403 

20   +  710 

+938 

-1438 

-282 

+  726   +878 

-1469 

—293   +  793 

+749 

—  1479 

—270 

21     +IO72 

+904 

—  1180 

-108 

+  1049 

+933 

—  1225 

—  135   +1126 

+821 

—  1260 

-116 

22     +I3S9 

+948 

-  820 

+  68 

+  1323 

+949 

-  880 

+  33   +1387 

+862 

—  940 

+  40 

23   |  +IS32 

+899 

—  399 

+242 

+  1501 

+927 

—  467 

+207 

+  1547 

+867 

—  544 

+202 

24   \  +I580 

+816 

+  54 

+405 

+  1572 

+863 

—  13 

+376   +1600 

+835 

—  IOI 

+359 

25   +1493 

+711 

+  494 

+551 

+  1525 

+765 

+  446 

+530   +1546 

+769 

+  355 

+509 

26   +1278 

+592 

+  885 

+677 

+  1353 

+632 

+  872 

+659  ;  +1379 

+667 

+  79' 

+644 

27 

+  957 

+466 

+  I2IO 

+778 

+  1070 

+479 

+  1221 

+759   +  1116 

+533 

+  "74 

+757 

28 

+  558 

+336 

+  1436 

+849 

+  70i 

+3i6 

+  1471 

+833  :  +  766 

+372 

+  1460 

+841 

29 

+  106 

+205 

+  1545 

+884 

+  271 

+  153 

+  I&00 

+874  <  +  353 

+  192 

+  1627 

+887 

30 

—  355 

+  73 

+  1534 

+883 

—  179 

—  i 

+  1609 

+886   -  84 

+  6 

+  I6S9 

+896 

3! 

—  791 

—  59 

+  1404 

+848 

-  621 

—  142 

+  1501 

+867   -  5i5 

-171 

+  1565 

+870 

32 

—  1164 

-183 

+  1169 

+78o 

—  1026 

-270 

+  1278 

+815   -  909 

-330 

+  1357 

+814 

33 

—1439 

—297 

+  861 

+690 

-1358 

-383 

+  96l 

+733   -1234 

-466 

+  1050 

+735 

34 

-1594 

-393 

+  497 

+58i 

—  1592 

—477 

+  574 

+621  :  —1476 

—573 

+  673 

+632 

35 

—1633 

—471 

+  134 

+465 

—  1704 

—550 

+  154 

+491  —1629  —652 

+  250 

+5" 

36 

—1560 

—537 

—  225 

+345 

-1684 

-602 

—  256 

+351   —  1661  !  —707 

-  183 

+370 

37 

—  1402 

—594 

-  544 

+226 

—  1549 

—630 

—  620 

+209  :  —1579   —733 

-  596 

+216 

38 

-1173 

-645 

-  818 

+  "5 

—  1317 

—639 

—  923 

+  77   —1384   —730 

—  959 

+  60 

39 

-  879 

-694 

-1058 

+  ii 

—  1017 

-637 

-1149 

—  43   —1098   —703 

—  1232 

-  85 

40 

—  540 

-740 

—1237 

-84 

—  674 

-631 

—  1301 

-150   -  752  :  -654 

—  1403 

—  219 

41 

-  159 

-783 

—  1347 

—  169 

—  303 

—627 

—244   —  375   —595 

—  1473 

-324 

42 

+  244 

-813 

—  1390 

-245 

+  80 

-626 

—324  1  +   3   —532 

—  1450 

—403 

43 

+  651 

-838 

-1365 

-315 

+  465 

-626 

—  1342 

—391  ;  +  3f>4   —477 

—  1357 

—462 

44 

+  1039 

-849 

-1275 

-383 

+  835 

—623 

—1225 

—442   +  697   —432 

—  1208 

—504 

45 

+  1399 

-845 

—  II2I 

—449 

+1181 

-617 

—  1049 

—478   +1001   —396 

—  1009 

—531 

46    +1708 

-821 

—  911 

—509 

+1482 

-607 

-  832 

—502  ;  +1264   —368 

-  778 

—542 

.17  '  +1946 

-768 

—  642 

-562 

+  1728 

-591 

-  580 

—517  :  +1480  —346 

—  520 

-536 

48    +2089 

-681 

—  322 

—60  1 

+  1908 

-566 

—  303 

—526   +1640   —328 

—  248 

—515 

49 

+2122 

—SV 

+  27 

—621 

+2014 

-524 

-  15 

—527   +1743   —312 

+  17 

-481 

50 

+2O26 

—393 

+  372 

-6n 

+2027 

—  460 

+  276 

—514   +1784  i  —298 

+  265 

—434 

51 

+  1803 

—207 

+  678 

—570 

+  1939 

-371 

+  548 

—479  i  +1761  i  —283 

+  487 

-378 

52 

+  1482 

—  ii 

+  912 

—499 

+  1744 

—254 

+  779 

—418  ,  +1679   —259 

+  673 

—314 

53 

+  1093 

+  178 

+  1054 

—396 

+  1454 

-116 

+  942 

—327    +1528  i  —221 

+  813 

-236 

54 

+  679 

+344 

+  1097 

—269 

+  1097 

+  27 

+  1017 

—213    +1313    —169 

+  895 

—145 

55 

+  279 

+477 

+  1042 

-117 

+  709 

+  166 

+  995 

—  80   +1044   —  ioi 

+  900 

—  3i 

56 

—  71 

+569 

+  906 

+  49 

+  33i 

+284 

+  884 

+  66   +743—27 

+  825 

+  IOO 

57 

—  344 

+612 

+  7io 

+217 

+   3 

+37i 

-  716 

+209  ;  +  451  i  +  43 

+  686 

+234 

58 

-  523 

+603 

+  497 

+370 

—  254 

+418 

+  536 

+337 

+  189   +99 

+  525 

+354 

59 

-  60  1 

+548 

+  312 

+489 

—  434 

+430 

+  385 

+437   -  27   +137 

+  397 

+437 

COEFFICIENTS  FOR  DIRECT  ACTION. 

TABLE   IVa. 
PERTURBATIONS  OF  THE  G-COORDINATE  X  OF  VENUS. 

The  tabular  unit  is  io-8. 


53 


Sys- 

tem 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

IO 

II 

i 

o 

+  47 

+  57 

+  92 

+  137 

+  181 

+217 

+218 

+184 

+142 

+107 

+  71 

+  52 

i 

+  37 

+  50 

+  81 

+118 

+  159 

+  197 

+212 

+  183 

+140 

+  92 

+  52 

+  30 

2 

—  ii 

+  ii 

+  41 

+  71 

+  109 

+  149 

+  176 

+  158 

+  114 

+  59 

+  8 

—  23 

3 

—  92 

—  59 

—  24 

+  5 

+  39 

+  75 

+  112 

•fill 

+  72 

+  9 

—  52 

-  97 

4 

—193 

—154 

—in 

—  79 

—  50 

—  14 

+  26 

+  46 

+  14 

-48 

—  122 

-183 

5 

—305 

-269 

—217 

-183 

-156 

—  122 

—  79 

-  41 

—  55 

—ill 

-191 

-268 

6 

—412 

—396 

-338 

—290 

-267 

-237 

—197 

—146 

-136 

—179 

—26l 

-348 

7 

—506 

—519 

—465 

—405 

-380 

—354 

-318 

-263 

-230 

—251 

-326 

-421 

8 

-581 

-628 

-586 

-487 

-468 

-437 

-384 

—331 

—330 

-386 

—482 

9 

-636 

-715 

-696 

—629 

-583 

-569 

—546 

-502 

-437 

—410 

—443 

—530 

IO 

-668 

-763 

-780 

-718 

-661 

—650 

-636 

-599 

—539 

-488 

—494 

-565 

ii 

—674 

—777 

-827 

-783 

-718 

-699 

-697 

—672 

—619 

—  556 

—534 

-582 

12 

-660 

—759 

-828 

—814 

—745 

—714 

-722 

—710 

-669 

—606 

-561 

-580 

13 

-621 

—710 

—790 

-803 

—745 

—699 

-707 

—712 

-682 

-625 

-569 

-560 

14 

—559 

-639 

-713 

-749 

-709 

-652 

—652 

-671 

—659 

—  610 

-552 

—523 

15 

—479 

—541 

—614 

-659 

—640 

-580 

—564 

-591 

-596 

-559 

—507 

-466 

16 

-383 

—431 

-495 

-542 

—540 

-487 

—452 

-475 

-496 

—476 

—432 

-389 

17 

—279 

-308 

—364 

—410 

-417 

-374 

-325 

—337 

-368 

—366 

—331 

-298 

18 

—  171 

-183 

-231 

—273 

-283 

-251 

—195 

-187 

—221 

—236 

-213 

-187 

19 

—  54 

—  61 

-98 

—  137 

—149 

—123 

—  68 

-  38 

—  66 

—  95 

—  84 

—  64 

20 

+  60 

+  56 

+  25 

—  13 

—  26 

__      T 

+  48 

+  93 

+  80 

+  46 

+  41 

+  58 

21 

+  170 

+160 

+  136 

+  100 

+  82 

+106 

+203 

+209 

+  179 

+  155 

+  171 

22 

+269 

+253 

+229 

+  197 

+  174 

+192 

+236 

+288 

+312 

+291 

+262 

+266 

23 

+350 

+330 

+305 

+275 

+249 

+259 

+299 

+350 

+384 

+377 

+348 

+342 

24 

+406 

+392 

+363 

+333   +307 

+306 

+341 

+387 

+430 

+434 

+412 

+398 

25 

+441 

+436 

+407 

+373   +347 

+338 

+367 

+412 

+451 

+466 

+456 

+437 

26 

+461 

+460  1  +435 

+400 

+374 

+361 

+378 

+418 

+457 

+475 

+476 

+464 

27 

+468 

+471 

+451 

+417 

+388 

+372 

+38o 

+418 

+454 

+478 

+483 

+478 

28 

+476 

+475 

+459 

+426 

+394 

+381 

+379 

+411 

+447 

+472 

+481 

+484 

29 

+480 

+475 

+463 

+432 

+397 

+381 

+379 

+405 

+440 

+465 

+475 

+485 

30 

+484 

+479   +467 

+438 

+401 

+380 

+380 

+401 

+434 

+459 

+473 

+479 

31 

+490 

+484   +476 

+445 

+406 

+380 

+38o 

+397 

+431 

+457 

+470 

+476 

32 

+491 

+492   +483 

+455  '  +414 

+380 

+379 

+392 

+426 

+456 

+469 

+473 

33 

+485 

+497   +488 

+460  i  +417   +379 

+375 

+387 

+419 

+451 

+467 

+466 

34 

+468 

+492 

+489 

+463   +420   +377 

+362 

+375 

+403 

+437 

+456 

+456 

35 

+439 

+468 

+479 

+458  \  +414   +366 

+341 

+354 

+377 

+4" 

+432 

+436 

36 

+397 

+426 

+448 

+437 

+392   +342 

+3io 

+314 

+338 

+368 

+391 

+399 

37 

+340 

+360 

+394 

+396 

+353 

+301 

+262 

+260 

+282 

4-311 

+331 

+343 

38 

+263 

+270 

+309 

+327 

+293 

+240 

+  196 

+  184 

+208 

+23"6 

+255 

+264 

39 

+  167 

+  166 

+  199 

+230 

+211 

+155 

+  109 

+  90 

+  112 

+141 

+161 

+166 

40 

+  56 

+  47 

+  69 

+  107 

+104 

+  5i 

+   2 

—  20 

—  4 

+  33 

+  55 

+  57 

41 

—  67 

-  80 

—  74 

-  38 

—  22    —  67 

—  121 

-145 

—127 

—  90 

—  59 

—  58 

42 

-188 

-206 

—215 

—  188  j  —  161   —192 

-254 

-283 

-265 

—222 

-180 

—172 

43 

—301 

-325 

—344 

—336 

-306 

—319 

-381 

—420 

-407 

—357 

—304 

—282 

44 

-397 

—430 

—459 

—464 

—443 

-438 

—494 

-549 

—543 

—486 

—423 

-385 

45 

—473 

—510 

—545 

-568 

-557   -544 

-589 

-656 

-665 

-609 

-534 

-475 

46 

-528 

-560 

-605 

—634 

-638   -626 

-655 

—729 

-760 

—712 

—628 

—555 

47 

-566 

-578 

-626 

-665 

—  681  i  -678 

—694 

—764 

-818 

—791 

—702 

-617 

48 

-585 

—574 

-611 

-660 

-685 

-692 

—703 

—759 

-833 

-833 

—749 

-657 

49 

-588 

-547 

-566 

—619 

-653 

-668 

—679 

-723 

-805 

-838 

-775 

-675 

50 

—572 

-508 

-506 

—549 

-588 

—609 

—627 

—659 

-737 

-796 

-766 

-668 

5i 

—539 

—457 

—431 

-454 

—499 

—525 

-545 

-571 

—639 

-716 

-723 

—639 

52 

—491 

—401   —347 

-350 

—391 

—423 

—447 

-470 

-525 

-605 

—644 

-589 

53 

—431 

—339   —268 

—248 

—276 

—3" 

—333 

-357 

—  401 

-477 

-535 

-518 

54 

-362 

—274   —193 

—152 

-162 

—197 

—  222 

—247 

-281 

-346 

-414 

—428 

55 

-283 

—205   —122 

-69 

-58 

-  88 

—  III 

—137 

-167 

—219 

-286 

—324 

56 

—109 

-134   -  58 

+   2 

+  31 

+  14 

—  14 

-  37 

-  64 

—  106 

—164 

—211 

57 

—114 

—  67   —  3 

+  60 

+  101 

+  102 

+  73 

+  48 

+  20 

—  14 

—  60 

—105 

58 

-  38 

-  8  ;  +  44 

+  104 

+151   +167 

+  144 

+  II5 

+  85 

+  54 

+  16 

—  18 

59 

+  19 

+  36  |  +  78 

+  130 

+179   +208 

+  194 

+  161 

+  128 

+  94 

+  59 

+  36 

•    .  .     fc       .  > 


INVESTIGATION  OF  INEQUALITIES  IN  THE 
MOTION  OF  THE  MOON  PRODUCED 
THE  ACTION  OF  THE  PLANETS 


BY 


SIMON  NEWCOMB 

ASSISTED    BY 

FRANK  E.  ROSS 


WASHINGTON,   D.    C.: 

PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OF  WASHINGTON 

JUNE,  1907 


Astron.  Oept. 


CARNEGIE   INSTITUTION   OF   WASHINGTON 


PUBLICATION  No.  72 


ASTRONOMY 


CONTENTS. 


INTRODUCTION I 

PART  I.  DEVELOPMENT  OF  THE  THEORY 3 

CHAPTER  I.  Fundamental  Differential  Equations 5 

1.  Notation  5 

2.  Dimensions  of  Quantities  in  Terms  of  Time,  Length,  and  Mass 6 

3.  Fundamental  Differential  Equations 6 

4.  Transformation  to  the  Moving  Ecliptic 6 

5.  Preliminary  Form  of  the  Potential  Function 8 

6.  Reduction  of  the  Terms  of  the  Potential  Function  for  the  Indirect  Action   9 

7.  Reduction  of  /?,  the  Potential  of  Direct  Action 10 

8.  Completed  Form  of  the  Fundamental  Equations n 

CHAPTER  II.  Development  and  Integration  of  the  Differential  Equations  for  the  Variation  of  the 

Elements 13 

9.  Fundamental  Variables 13 

10.  Canonical  Form  of  the  Differential  Equations 14 

n.  Transformation  of  the  Canonical  Elements 14 

12.  Form  of  the  Partial  Derivatives 15 

13.  Numerical  Values  of  the  Fundamental  Quantities 15 

14.  Formation  of  the  Transformed  Differential  Equations 16 

15.  Elimination  of  the  Time  in  Certain  Cases 19 

CHAPTER  III.  Definitive  Form  of  the  Differential  Variations  of  the  Elements 21 

16.  General  View  of  the  Problem 21 

17.  Reduction  of  the  Equations  for  the  Direct  Action 21 

18.  Notation  of  the  Planetary  Factors 23 

19.  Notation  of  the  Lunar  Factors 23 

20.  Numerical  Form  of  the  Fundamental  Coefficients 24 

21.  Fundamental  Differential  Equations  for  the  Direct  Action 26 

22.  Reduction  of  the  Equations  for  o,  e,  and  y  to  Numbers 26 

23.  Reduction  of  the  Equations  for  /,  IT,  and  6  to  Numbers 26 

24.  Development  of  the  Indirect  Action 28 

25.  Abbreviated  Coefficients  for  the  Indirect  Action 30 

26.  Integration  of  the  General  Equations 31 

27-  Inequalities  of  /,  T,  and  9 32 

28.  Treatment  of  the  Non-periodic  Terms  in  /? 33 

29.  Adjustment  of  the  Arbitrary  Constants 35 

30.  Opposite  Secular  Effects  of  the  Direct  and  Indirect  Action  of  a  Planet  near  the  Sun 35 

PART  II.  DEVELOPMENT  OF  THE  PLANETARY  COEFFICIENTS 37 

CHAPTER  IV.   Coefficients  for  the  Direct  Action 39 

31.  Remarks  on  the  Method  of  Development  by  Mechanical  Quadratures 39 

32.  Action  of  Venus,  Systems  of  Coordinates 41 

33.  Action  of  Venus,  Fundamental  Data  for  the  -^-coefficients 42 

34.  Explanation  of  Tables  of  ^-coefficients  for  Venus 44 

35.  Mechanical  Development  in  a  Double  Periodic  Series., , , ,, 46 

36O5S8  m 


PREFACE. 

THE  immediate  incentive  to  the  present  work  was  the  hope  of  explaining  by 
gravitational  theory  the  observed  variations  in  the  mean  longitude  of  the  Moon, 
shown  by  more  than  two  centuries  of  observation  to  exist,  but  not  yet  satisfactorily 
accounted  for.  The  author  has  published  a  number  of  papers  and  memoirs  on  this 
subject  during  the  last  forty  years,  terminating  with  a  summary  of  the  case,  which 
appeared  in  the  Monthly  Notices  of  the  Royal  Astronomical  Society  for  March, 
1904.  The  deviations  in  question  offer  the  greatest  enigma  yet  encountered  in 
explaining  the  motions  of  the  heavenly  bodies,  and  the  present  paper  may  be 
regarded  as  a  contribution  to  the  study  of  the  problem  thus  offered. 

While  the  work  was  in  progress  the  completing  chapter  of  Professor  Brown's 
Theory  of  the  Moorfs  Motion  appeared.  The  actual  work  being  based  on  De- 
launay's  theory,  it  seemed  to  be  desirable  to  revise  and  correct  it  by  Brown's 
results.  In  doing  this  the  imperfections  of  Delaunay's  theory  as  a  basis  became  so 
evident,  and  the  later  theory  proved  to  be  so  much  better  adapted  to  the  purpose 
of  the  investigation,  that  the  completed  work  gradually  became  step  by  step  prac- 
tically based  upon  Brown's  theory,  except  in  those  parts  requiring  derivatives 
which  could  not  be  readily  obtained  except  from  Delaunay's  literal  expressions. 
Acknowledgment  is  due  to  Professor  Brown  for  courteous  advice  and  assistance 
which  facilitated  the  use  of  his  work  for  the  purpose. 

The  theory  of  the  action  of  the  planets  on  the  Moon  being,  in  several  points,  the 
most  intricate  with  which  the  mathematical  astronomer  has  to  deal,  it  is  important 
that  its  development  should  be  presented  in  a  form  to  render  as  easy  as  possible  the 
detection  of  errors  or  imperfections.  In  the  arrangement  of  the  work  this  end  has 
been  kept  constantly  in  view.  It  is  hoped  that  any  investigator  desiring  to  test  the 
processes  will  find  few  difficulties  except  those  necessarily  inherent  in  the  nature 
of  the  work. 

To  form  a  general  conception  of  the  arrangement  it  may  be  stated  that  the  work 
naturally  divides  itself  into  four  parts.  One  of  these  treats  of  the  theory  of  the 
subject,  including  under  this  head  not  only  the  general  equations,  but  the  numerical 
details  on  which  all  the  computations  are  based.  In  this  part  the  fundamental 
quantities  are  reduced  to  products  of  two  factors,  one  of  which  depends  upon  the 
coordinates  of  the  planet;  the  other  upon  the  geocentric  coordinates  of  the  Moon. 
The  first  factors,  termed  planetary,  are  numerically  developed  in  Part  II.  This 
development  falls  into  two  parts,  one  treating  the  direct  action  of  the  planet,  the 
other  the  indirect  action  through  the  Sun.  In  Part  III  is  found  the  numerical 

VII 


VIII  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

development  of  the  factors  depending  upon  the  Moon  alone,  and  of  their  partial 
derivatives  as  to  the  lunar  elements.  In  Part  IV  is  presented  the  combinations  of 
these  two  factors  and  the  final  results  of  the  work. 

A  more  complete  summary  in  detail  is  found  in  the  table  of  contents.  An  effort 
has  been  made  to  lessen  the  trouble  of  finding  the  definitions  of  the  symbols  used 
by  collecting  in  the  introduction  definitions  or  references  to  these  symbols  as  to 
the  meaning  of  which  doubt  might  be  felt. 

A  word  may  be  added  as  to  the  part  taken  by  the  author's  assistant.  At  an 
early  stage  in  the  work  Dr.  Ross  made  a  practically  independent  computation  of 
the  principal  periodic  inequalities,  using  the  methods  of  Hill  and  Radau.  In  doing 
this  he  discovered  the  error  of  the  Jovian  evection  as  computed  by  them,  which 
arose  from  the  omission  of  what  we  may  call  the  side-terms  in  the  indirect  action. 
His  result  for  the  coefficient  was  i".i6,  in  exact  agreement  with  that  originally 
found  by  Mr.  Neville.  In  this  early  stage  of  the  work  the  writer  did  not  intend  to 
do  much  more  than  revise  these  computations,  and  make  a  thorough  investigation 
of  the  terms  of  long  period.  But  he  found  the  theory  of  the  subject  so  interestingj 
and  the  opportunity  for  recasting  the  methods  so  attractive,  that  he  was  led  to 
carry  the  work  through,  with  Dr.  Ross's  assistance,  on  the  basis  of  his  own 
developments. 

The  next  step  in  logical  order  is  the  rediscussion  of  the  moon's  mean  longitude 
since  1650,  as  derived  from  occultations  of  stars,  with  a  view  of  learning  what 
modifications  will  be  produced  by  the  use  of  the  more  rigorous  data  now  available, 
and  the  addition  of  thirty  years  to  the  period  of  available  observations.  This  redis- 
cussion will,  the  writer  hopes,  be  his  next  contribution  to  the  subject  of  the  motion 
of  the  Moon. 

It  remains  to  add  that  the  work  has  been  prosecuted  under  the  auspices  of  the 
Carnegie  Institution  of  Washington,  without  the  help  of  which  it  could  not  have 
been  undertaken. 

SIMON  NEWCOMB. 
WASHINGTON,  MAY,  1907. 


ACTION   OF  THE  PLANETS  ON  THE  MOON. 

INTRODUCTION. 

MORE  than  thirty  years  ago  the  author  proposed  to  treat  the  action  of  the 
planets  on  the  Moon  by  using  the  Lagrangian  differential  equations  for  the  variation 
of  the  elements  by  considering  as  simultaneously  variable,  not  only  what  are  com- 
monly called  the  elements  of  the  Moon,  but  those  of  the  orbit  of  the  centre  of  mass 
of  the  Earth-Moon  around  the  Sun  also.*  Twelve  elements  would  thus  come  in, 
and  the  coordinates  both  of  the  Moon  and  of  the  Sun  would  be  expressed  in  terms 
of  the  osculating  values  of  all  these  elements. 

Notwithstanding  the  favorable  opinion  of  this  method  expressed  at  the  time  by- 
Professor  Cayley,  and  later,  as  to  some  of  its  processes,  by  Professor  E.  W.  Brown, 
the  author  found  that,  in  applying  it  unmodified,  which  he  did  during  the  years 
1872-77,  very  long  and  complex  computations  were  required  in  its  application. 
The  result  was  that  the  work,  so  far  as  it  was  carried,  remained  unpublished  for 
nearly  twenty  years.  Hoping  that  the  general  developments  of  the  work  and  some 
of  the  details  might  be  of  use  to  subsequent  investigators,  the  incomplete  work 
was  finally  published  in  1895. 

About  the  same  time  with  the  publication  of  this  work  appeared  the  very  elabo- 
rate one  of  Radau.  f  This  work  contains  a  seemingly  exhaustive  enumeration  of 
possible  inequalities  of  long  period,  and  the  numerical  computation  of  a  great  num- 
ber of  lunar  inequalities  due  to  the  action  of  the  planets  which  had  not  previously 
been  suspected. 

On  recommencing  the  work  in  1904  it  became  very  clear  to  the  author  that  its 
completion  by  his  former  method,  unmodified,  would  be  impracticable,  and  that 
satisfactory  results  could  best  be  reached  by  regarding  the  solar  elements  as  con- 
stants, or  known  variables  from  the  beginning.  In  the  present  investigation,  there- 
fore, the  method  has  been  modified  so  that  the  final  values  of  the  coordinates  of 
the  Moon,  instead  of  being  expressed  as  functions  of  the  instantaneous  elements 
of  the  Earth's  disturbed  motion,  are  expressed  as  functions  of  the  mean  elements. 
As  thus  modified  it  is  substantially  a  continuation  of  that  of  Delaunay,  as  applied 

*  Liouv Hie,  Journal  des  Mathematiques,  1871,  March. 

t  Annales  tie  I' Observatoire  de  Paris,  Me'mofres,  vol.  XXI. 


2  INTRODUCTION. 

first  by  Hill  and  then  by  Radau.  In  this  method  the  coordinates  of  the  Sun, 
relative  to  the  centre  of  gravity  of  the  Earth  and  Moon,  are  regarded  as  known 
functions  of  the  time.  Then,  when  the  action  of  the  Sun  alone  is  considered, 
the  coordinates  of  the  .Moon  relative  to  the  Earth  are  found  by  the  method  of 
Deteiin2y^>c9ftiijBet^d.if  necessary,  as  functions  of  six  purely  arbitrary  constants. 

This  solution  of  the  problem  of  three  bodies  is  supposed  to  be  complete  in 
advance.  When  the  action  of  the  planets  is  then  taken  into  consideration,  the 
only  elements  whose  variations  are  to  be  determined  by  the  Lagrangian  equations 
are  the  six  final  elements  of  the  Moon's  motion.  The  variations  in  the  coordinates 
of  the  Sun,  due  to  the  same  action,  are  derived  with  great  ease,  and  enter  into  the 
differential  equations.  In  this  way  a  system  of  six  differential  equations  for  the 
determination  of  the  changes  in  the  lunar  elements  is  all  that  is  necessary. 

In  setting  forth  the  subject  it  is  deemed  unnecessary  to  repeat  the  derivation  of 
the  equations  already  found  in  astronomical  literature.  For  this  branch  of  the 
subject,  reference  may  be  had  to  Hill's  paper  in  the  American  Journal  of  Mathe- 
matics, Vol.  VI,  and  to  Chapter  XIII  of  the  Treatise  on  the  Lunar  Theory  by 
E.  W.  Brown.  It  is  deemed  necessary  only  to  explain  fully,  at  each  point,  the 
application  of  the  method,  and  the  meaning  of  the  symbols  introduced. 


PART  I. 

DEVELOPMENT  OF  THE  THEORY. 


CHAPTER    I. 

FUNDAMENTAL   DIFFERENTIAL   EQUATIONS. 

§  i.  Notation.     The  following  notation  is  mostly  used  in  this  work: 
G,  when  designating  a  point,  centre  of  mass  of  Earth  and  Moon;  m',  mass  of  the 
Sun;  ;«2,  mass  of  the  Earth;  /«3,  mass  of  the  Moon;  m4,  mass  of  the  Planet. 

H  =  m2  +  m3  fj.'  —  m'  +  /* 

x,  y,  z,  r,  geocentric  coordinates  and  radius  vector  of  the  Moon,  referred 
to  the  moving  ecliptic; 

x',  y',  z',  r',  coordinates  and  radius  vector  of  the  Sun,  referred  to  the  point 
G  and  the  moving  ecliptic; 

£,  17,  £,  and  p,  the  ratios  of  x,  j,  z,  and  r  of  the  Moon  to  the  mean  dis- 
tance of  the  latter:  x  =  at;,  etc.  When  unmarked  the  coordinates 
are  referred  to  a  moving  J^-axis  directed  toward  the  mean  Sun; 

#1,  jj,  Moon  coordinates  referred  to  the  mean  Moon  as  the  Jf-axis; 

A,  distance  of  the  Planet  from  G; 

S,  cosine  of  angle  between  rand  r'\ 

S',  cosine  of  angle  between  r  and  A  ; 

/>(,,  potential  function  of  mutual  action  of  Earth  and  Moon; 

11,  potential  function  for  action  of  Sun  on  Moon; 

7?,  potential  function  for  action  of  Planet  on  Moon; 

/,  TT,  6,  mean  longitude,  longitude  of  perigee,  longitude  of  node  of  Moon; 

TTj,  #!,  motions  of  IT  and  0  in  unit  of  time  (quantities  of  dimensions  T~l) ; 

N,  motion  of  argument  in  unit  of  time; 

n,  ratio  of  motion  of  an  argument  to  n,  the  mean  motion  of  the  Moon; 

v,  the  integrating  factor,  generally  =W/N; 

a,  e,  g,  defined  in  (43),  §22; 

K>  C,  D,  planetary  coefficients  for  the  direct  action,  defined  in  §  20; 

p,  y,  K4,  lunar  coefficients,  §  20  Eq.  (36) ; 

6r,y,  /,  planetary  coefficients  for  the  indirect  action,  defined  in  §  24; 

G  is  also  used  for  a  combined  lunar  and  planetary  argument; 

a,  logarithm  of  a,  the  Moon's  mean  distance; 

v,  M,  j,  s,  the  mean  longitudes  of  the  respective  planets,  Venus,  Mars, 
Jupiter,  and  Saturn  measured,  in  each  case,  from  the  Earth's  peri- 
helion: TT'  for  i8oo  =  99.°5. 

5 


6  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

The  abbreviation  "  Action  "  has  been  used  to  designate  the  previous  work  of 
the  author  on  this  subject  —  "Theory  of  the  Inequalities  in  the  motion  of  the  Moon 
produced  by  the  Action  of  the  Planets";  forming  Part  III  of  Astronomical  Papers 
of  the  American  Ephemeris,  Vol.  V. 

§  2.  Dimensions  of  quantities.  In  this  subject  it  will  be  found  helpful  to  the 
reader  and  investigator  to  have,  in  the  case  of  the  principal  equations,  a  statement 
of  their  dimensions  in  terms  of  the  fundamental  units  of  Mass,  Time,  and  Length. 
In  strictness  an  independent  unit  of  mass  is  not  necessary  in  gravitational  astronomy, 
because  the  most  convenient  unit  is  that  mass  which,  on  an  equal  mass  at  unit 
distance,  exerts  a  unit  force  of  gravitation.  But  it  is  still  sometimes  convenient  to 
use  this  unit  in  the  equations,  although  it  is  a  derived  one. 

In  the  case  of  each  system  of  equations  which  are  regarded  as  fundamental  will 
be  found  the  dimensions  of  the  terms  which  form  its  members,  the  signification 

being  as  follows: 

T,  Time  ;         Z,  Length  ;         M,  Mass. 

The  definition  of  the  unit  of  mass  just  given  leads  to  the  relation 


In  this  way  it  will  be  much  easier  than  it  would  be  without  this  help  to  appreciate 
the  degree  of  magnitude  of  small  quantities.  Considered  by  itself,  no  concrete 
quantity  can  be  regarded  as  small  or  great;  it  is  so  only  when  compared  with  other 
quantities  of  the  same  kind,  or,  to  speak  more  accurately,  of  the  same  dimensions 
in  fundamental  quantities.  The  ratios  of  two  fundamental  quantities  of  the  same 
kind  are  pure  numbers,  and  these  may  be  large  or  small  to  any  extent. 

§  3.    Fundamental  differential  equations. 
Putting 

x\t  y»  z\i  tne  geocentric  coordinates  of  the  Moon  referred  to  any  system 

of  fixed  axes, 
P,  the  total  potential 

the  differential  equations  to  be  integrated  may  be  written 

dP  dP  dP 


-  [  Dimensions  =  AfZ.-'  =  LT~*1       (j) 

dxl  dyl  ozl 

§  4.  Transformation  to  the  moving'  ecliptic.  In  the  preceding  equations  the 
coordinates  are  referred  to  fixed  axes.  In  astronomical  practice  the  coordinates 
of  the  heavenly  bodies  are  referred  to  the  moving  ecliptic.  The  latter  carries  the 
plane  of  the  Moon's  orbit  with  it  in  its  motion.  It  therefore  seems  desirable  to 
refer  the  motion,  in  the  first  place,  to  the  moving  ecliptic. 


FUNDAMENTAL  DIFFERENTIAL   EQUATIONS.  7 

To  do  this  let  us  put 

x,  y,  z,  coordinates  referred  to  the  moving  ecliptic; 
K,  the  speed  of  motion  of  the  plane  of  the  ecliptic; 

II,  the  longitude  of  the  ascending  node  of  the  moving  on  the  fixed  ecliptic, 
or  of  the  instantaneous  axis  of  rotation  of  the  ecliptic.  At  the 
present  time  we  have  II  =  173°,  nearly. 

Then,  regarding  nt  as  infinitesimal,  the  expression  for  the  moving  coordinates  in 
terms  of  the  fixed  ones  will  be 

x  =  .*•,  —  zjic  sin  II 
y  =  yl  +  Z^K  cos  II 

z  =  zl  +  x^tic  sin  II  —  y^K  cos  13 
Putting  for  brevity 

p  =  K  sin  II  q  =  K  COS  II  [Dim.  of  /,  y,  and«=  T~l  ], 

these  expressions  become 


(2) 

Differentiating  them  twice  as  to  the  time,  regarding  p  and  q  as  constant,  we  have 
Dfx  =  Dfx,-ptDfzl-2pD^ 

?y,  +  qtDfz,  +  2qDtz,  (3) 


Dfz  =  Df 

Regarding  P,  originally  a  function  of  xi9  yly  and  zlt  as  becoming  a  function  of*,  y, 
and  z  through  the  substitution  (2)  we  have 

dP      dP         dP  dP      dP         dP  dP     dP          dP         dP 

dx~=='6X+  **&  =~d~  qtte  aF  "  dz  ~^dx  +  qtS 


Substituting  these  expressions  for  D\x^  D\y^  and  DIZ^  in  (3)  and  dropping  terms 
of  the  second  order  in  pt  and  qt  we  find 

dP 


™  +  *qDfl  (4) 


dP 

D?z=-B-z+ 

Equations  of  this  form  were  used  by  Hill  for  the  same  purpose. 

*  Annalfyf  Maihematict,  vol.  I,  1890, 


8  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

It  follows  that  if  we  add  to  P  the  terms 

A*  =  2p(zD^  -  *Dfl)  +  2q(yD^  -  zDty,}  (5) 

so  that  the  potential  shall  become 

P  +  LR  [Dim.  =  3/z,-'  =  z,"  r-'] 

the  fundamental  differential  equations  in  x,  y,  and  z,  will  retain  the  form  (i) 
unchanged,  and  the  coordinates  referred  to  the  moving  ecliptic  will  be  determined 
by  the  general  equations 

dP  dP  BP 

£>'x  =  -f  D?y  =  if-  Dt2z  =  -£•  (6) 

dx  dy  dz 

In  A/?  the  symbols  x1}  y1}  and  zi  have  the  same  meanings  as  x,  y,  and  z,  but  they 
are  to  be  regarded  as  constant  when  AT?  is  differentiated  as  to  the  lunar  elements. 

§  5.    Preliminary  form  of  the  potential  function. 

We  put  fl  for  the  part  of  the  potential  P  due  to  the  action  of  the  Sun.  This 
part  is  developed  in  a  series  proceeding  according  to  the  powers  of  r\r'  in  the 
well-known  form 


where  S,  the  cosine  of  the  angle  between  the  radii  vectores  of  the  Moon  and  Sun 
from  the  point  G,  is  determined  by  the  equation 

rr'S  =  xx'  +  yy'  +  zz' 

When  we  assign  to  x',  y',  z',  and   r1  their  elliptic  values,  we  have  what  may  be 
called  the  Delaunay  part  of  the  potential.     We  put 

f!0,  the  Delaunay  part  of  fi. 

Op,  the  increment  of  fl0  produced  by  the  action  of  the  planets  on  the  Earth. 

The  part  /?  of  P,  due  to  the  direct  action  of  the  planet  in  changing  the  coordi- 
nates of  the  Earth  relative  to  the  Moon,  may  be  formed  from  fl  in  (7)  by  replacing 

m'  '  ,  r'  ,  x',  y',  and  z' 

by 

mt,  A,  X,  Y,  and  Z 

where  ;«.,  is  the  mass  of  the  planet,  and  A,  X,  K,  and  Z  its  distance  and  coordi- 
nates relative  to  the  point  G.     Putting  R  for  this  part  we  have  for  its  principal  term 


FUNDAMENTAL  DIFFERENTIAL  EQUATIONS.  9 

where  S'  is  determined  by  the  equation 

r*S'  =  (*'  +  xjx  +  (y'  +  yjy  +  «+  z^z 

•*4>  J«  and  ^4  being  the  heliocentric  coordinates  of  the  planet. 

We  have  thus  separated  the  potential  of  all  the  actions  changing  the  coordinates 
of  the  Moon  relative  to  the  Earth  into  the  following  five  parts. 

A.  The  part  generated  by  the  mutual  action  of  the  Earth  and  Moon,  />„  =  /*/  r, 
which  taken  alone  would  give  rise  to  an  undisturbed  elliptic  motion  of  the  Moon 
around  the  Earth. 

B.  The  part  fig  generated  by  the  action  of  the  Sun,  assuming  the  point  G  to  move 
in  an  elliptic  orbit. 

C.  The  part  ftp,  the  increment  of  ft0  due  to  the  action  of  the  planets  on  the 
point  G. 

D.  The  part  /?  due  to  the  direct  action  of  the  planet.     Developed  in  the  same 
way  as  the  highest  term  of  ft  the  principal  term  of  this  part  is  formed  from  SI  by 
replacing  m',  *•',  y',  and  z'  by  the  mass  and  G-coordinates  of  the  planet.     The  value 
of  its  principal  term  is  given  in  (7«). 

E.  The  part  A/?  arising  from  the  reference  of  the  coordinates  to  the  moving 
ecliptic. 

The  complete  value  of  P  thus  becomes 

p=  pe  +  n0  +  np  +  j?  +  A/?  (8) 

and  we  are  to  consider  this  expression  as  replacing  P  in  the  equations  (6). 

§  6.  Reduction  of  the  terms  of  the  potential  function  for  the  indirect  action. 

By  substituting  tor  S  in  (7)  its  value,  the  first  and  principal  term  of  ft  becomes  a 
linear  Junction  of  the  six  squares  and  products  of  the  lunar  coordinates  *•,  y,  and  z, 
which  we  may  write 

ft  =  7>J  +  TJ  +  7X  +  *  Ttxy  +  a  7>*  +  2  Ttyz  (9) 


Moreover,  since  we  form  the  part  ftp  of  the  potential  by  assigning  increments  to 
T,  and  the  part  R  by  making  T  &  function  of  the  elements  of  the  planet,  it  follows 
that  both  of  these  parts  as  well  as  ft  are  of  this  same  form. 

For  the  first  and  principal  term  of  ft0  in  which  the  higher  powers  of  rjr'  are 
dropped  we  have 


-^—  x'y> 


-,-,--  -.  .  (10) 

r'3\2  r>*      2J  2  r'*    r'* 

—  (S—   -\  T  -  3  m>  y'z' 

r's\2  r>>      2)  2'~  2  r>*'   r>* 


io  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

The  study  of  the  second  term,  which  it  may  be  advisable  to  examine  for  sensible 
results,  is  postponed,  and  ft  is  taken  as  equal  to  its  principal  part.  The  value  of  flp 
is  then  found  by  adding  to  the  preceding  values  of  Tt  their  increments  produced 
by  the  action  of  the  planets  upon  the  coordinates  #',  y'  ,  and  z'  of  the  Sun.  If  we 

put 

v',  the  longitude  of  the  Sun 

and   take   the  moving  ecliptic  as   the  plane  of  reference,  we   may  regard  z',  the 
periodic  perturbations  of  the  latitude,  as  infinitesimal  and  write 

x'  =  r'  cos  v'  y'  =  r'  sin  v'  z'  =  r'  sin  /S' 

where  /3'  is  the  Sun's  latitude,  a  minute  purely  periodic  quantity. 

Substituting  these  values  in  (io),  the  expressions  for  the  coefficients  T  become 

T,  -  ^,(t  +  f  cos  2»')  Tt  =  ~  (i  -  |  cos  2*')  T*=-~» 

(LOO) 

„      3  m'    .  3  ;«'  sin  /3'  cos  z/  ~      3  »''  sin  /3'  sin  r' 

y,  =  --  s  sin  2z»'  /   =- 


r 


i  —  =  --  5 

3  2  r'3 


If  we  assign  to  these  quantities  their  elliptic  values,  (7)  will  become  fl0  for  which 
the  integration  is  assumed  in  advance.  We  have  now  to  assign  to  v'  and  /'  the 
increments  8v'  and  r'8p',  p'  being  the  Naperian  logarithm  of  r'  .  The  resulting 
increments  of  the  coefficients  are 

87;=  ^j  {-2  sin  2w'8»'-3  cos  2v'&p'-&p'}    87>=  ^73  {2  sin  2^'Sz;'  +  3  cos  2»'8/»'-V} 

(ii) 
5  r3=  —-3  V  8  Tt=  ^3  {  2  cos  2r'Sw'  -  3  sin  iv'Bp'  } 

2^*  4^* 

The  values  of  8T5  and  87"6  will  be  the  original  values  (9)  of  7"5  and  T6  as  they 
are  due  wholly  to  the  action  of  the  planet.  With  them  the  expression  for  flp 
derived  from  (9)  becomes 


flf  =  8  7>2  +  8  T2y2  +  8  7>»  +  28  7>j  +  2  7>*  +  2  7^*  (12) 

§  7.     Reduction  of  R,  the  potential  of  direct  action. 

By  substituting  for  S'  in  the  principal  term  (7«)  of  ./?  its  expression  in  terms  of 
the  G-coordinates  of  the  planet  we  shall  have 


where 

-f  Cz2  -f  2/?*_y  -f  lExz  +  iFyz  [Dim.  =  z,-' 


FUNDAMENTAL  DIFFERENTIAL  EQUATIONS.  n 

the  values  of  the  coefficients  being 

(*'  +  *4)2     i  i  n     (*'  + 

—-    ~3 


[Dun.  =  Z,  •]     (14) 


3  A3 


It  should  be  noted  that  these  coefficients  require  the  factor  f  to  make  them  directly 
comparable  with  ZJ,  T2,  etc.,  in  (10). 

§  8.   Complete  form  of  the  fundamental  equations. 

Comparing  the  expressions  (12)  to  (14)  we  see  that  flp  and  R  are  of  the 
same  form,  and  that  the  principal  terms  of  each  are  products  of  two  factors,  of 
which  one  depends  solely  on  the  heliocentric  coordinates  of  the  Sun  and  planet, 
and  the  other  is  a  square  or  product  of  the  coordinates  of  the  Moon.  Moreover,  if 
we  put,  for  brevity, 

/»  =  o,  +  R  +  A;?  (is) 

the  fundamental  differential  equations  may  be  written 


where  x,  y,  and  z  are  coordinates  referred  to  the  moving  ecliptic  as  the  fundamental 
plane. 

We  shall  now  consider  these  differential  equations  as  solved  for  the  case  when 
PI  is  dropped  from  the  second  members.  The  problem  will  then  be  that  of  the 
solution  when  Pt  is  included;  and  this  problem  will  be  attacked  by  the  Lagrangian 
method  of  variation  of  elements. 


CHAPTER    II. 

DEVELOPMENT  AND  INTEGRATION  OF  THE  DIFFERENTIAL  EQUATIONS 
FOR  THE    VARIATION   OF  THE   ELEMENTS. 

§  9.  The  problem  being  to  integrate  equations  (16),  we  shall  regard  as  known 
quantities  the  coordinates  x',  y',  z'  of  the  Sun,  which  enter  implicitly  into  the  equa- 
tions, as  well  as  those  of  the  planets  relative  to  the  Sun.  The  problem  then  is  to 
express  the  values  of  x,  y,  and  z  in  terms  of  the  fundamental  constants  implicitly 
contained  in  the  differential  equations,  and  six  other  arbitrary  constants  which  we 
regard  as  elements  of  the  Moon's  motion. 

The  solution  of  the  equations  is  separated  into  two  parts  by  applying  the  La- 
grangian  method  of  the  variation  of  elements.  We  have  first  the  Delaunay  solu- 
tion, in  which  Pt  is  dropped.  This  solution  gives  the  orbit  of  the  Moon  around 
the  Earth  under  the  influence  of  the  Sun's  and  Earth's  attraction  alone.  From  it 
we  are  to  pass,  by  the  method  of  variation  of  elements,  to  a  solution  when  P±  is 
taken  account  of. 

We  accept  the  results  of  Delaunay,  as  found  in  his  work,  as  forming  the  basis  of 
the  first  solution,  the  results  needing  only  certain  modifications  in  the  terms  depend- 
ing on  the  Sun's  parallax,  arising  from  the  tact  that  he  did  not  take  into  account 
the  mass  of  the  Moon,  and  certain  reductions,  to  reduce  them  to  the  required  form. 
This  being  done  we  have  values  of  the  Moon's  coordinates  satisfying  the  differential 
equations  in  the  case  P=p/r-\-tl0  and  expressed  as  functions  of  six  arbitrary 
constants 

«»  c,  7»  4)'  "o>  ^o 

and  of  the  time  /.     The  latter  enters  only  through  the  quantities  /,  IT,  and  0,  named 
and  defined  thus 


Mean  longitude  :  l—t^+nt     Long,  of  perigee  :  TT—Tr^+irJ     Long,  of  node  :  0=00+fy  (17) 

where  n,  irt,  and  0,  are  functions  of  a,  e,  and  y. 

I  use  the  quantities  7,  IT,  and  6  instead  of  Delaunay's  /,  g,  and  h,  which  are  the 
mean  anomaly,  the  angle  node  to  perigee,  and  the  longitude  of  the  node.  The 
expressions  for  the  symbols  used  here  in  terms  of  those  used  by  Delaunay  are 
therefore 

/  =3  Delaunay's  //  +  g  +  J         TT  =  Delaunay's  h  +  g         6  =  Delaunay's  h         (18) 

13 


14  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

The  fundamental  idea  of  the  Lagrangian  method,  which  we  propose  to  apply 
to  the  present  problem,  is  that  the  six  arbitrary  elements  are  to  become  such 
functions  of  the  time  that  the  solution  which  satisfies  (16)  when  Pt  =  o  shall  still 
satisfy  it  when  the  variable  values  of  the  elements  are  substituted  for  the  constant 
values  in  the  expressions  for  the  coordinates.  The  derivatives  of  the  elements  as  to 
the  time  may  be  formed  by  known  processes,  but  the  details  of  these  processes  are 
unnecessary,  because  Delaunay  gives  their  results  in  a  form  most  convenient  for 
our  purpose. 

§  10.   Canonical  form  of  the  differential  equations. 

We  see  from  (5),  (12),  (13),  and  (15)  thatPt  is  a  function  of  given  quantities  and  of 
the  Moon's  coordinates.  By  substituting  for  the  latter  their  expressions  in  terms  of 
the  six  arbitrary  constants  of  the  first  integration,  /\  becomes  a  function  of  a,  e,  y,  /, 
TT,  and  0.  The  differential  variations  of  the  elements  are  then  expressed  in  the  most 
condensed  form  by  replacing  a,  e,  and  y  by  three  other  quantities  clf  c2,  and  c3,  func- 
tions of  a,  e,  y,  so  chosen  that  the  differential  equations  to  be  solved  shall  be 


(19) 


The  variable  elements  ca,  c2,  and  c3  are  functions  of  Delaunay's  Z,  G,  II. 

CI  =  L  C,=  G-L  CS  =  H-G  (20) 


[Dim.  =  L\M*  = 

§  ii.    Transformation  of  the  canonical  elements. 

The  canonical  elements  cw  c2,  and  cs  can  not  be  used  explicitly  in  the  processes 
of  solution.  We  have  therefore  to  express  them  in  terms  of  a,  e,  and  y.  The 
values  of  Z,  £?,  and  H  are  not  given  by  Delaunay  in  terms  of  the  final  a,  e,  and  y, 
but  of  preliminary  ones  from  which  the  required  expressions  are  to  be  derived  as 
follows  : 

1.  In  Vol.  II,  pp.  235-236,  Delaunay  gives  the  expressions  for  Z,  G,  and  //  in 
terms   of  the    a,  e,   and  y  which    resulted    immediately  from    his    processes  of 
integration. 

2.  On  p.  800  he  gives  the  transformation  of  these  a,  e,  y,  into  the  final  values  of 
these  quantities  which  appear  in  the  expression  for  the  Moon's  coordinates,  which 
are  those  we  are  to  use. 


DEVELOPMENT  OF  THE  DIFFERENTIAL  EQUATIONS.  15 

To  find  from  these  data  the  expressions  for  the  derivatives  of  Z,  G,  H  in  terms 
of  the  final  a,  e,  y,  I  shall  write  a,  e,  g,  n,  for  the  quantities  a,  e,  y,  »,  as  found  on 
pp.  235-236  of  Delaunay,  Vol.  II,  and  shall  also  put 

n' 

m  =  - 
n 

The  forms  which  we  have  to  use  are: 

L,  G,  Jf=/(a,  e,  g,  m)  a,  e,  g  =/(«,  e,  7,  »z)  (21) 

Noticing  that  m  is  a  function  of  a  and  m  of  a,  we  shall  then  have 
dL      (dL      dLdm\da      dL  de      dL  dg 

^  I   i    I i    i __o.  /  2  2  ^ 

da       \  da      dm  da  J  da       de  da      dg   da 
with  similar  forms  for  G  and  H. 

§  12.  Form  of  the  partial  derivatives.  Two  points  in  the  use  of  the  partial 
derivatives  are  these: 

a.  In  taking  the  partial  derivatives  I  use  the  logarithm  of  a  and  of  a  instead  of 
these  quantities  as  the  variables  with  respect  to  which  derivatives  are  to  be  formed. 
Homogeneity  in  the  equations  is  thus  secured,  the  variables  being  all  pure  numbers, 
or  quantities  of  dimensions  o.  We  put 

a  —  log  a  whence  a  =  ea 

ft.  The  quantities  n  and  n  are  defined  as  functions  of  a  and  of  a  respectively  by 

the  equations 

a'n2  =  oW  =  p 

It  follows  that  if  we  have  an  expression  M  developed  in  powers  of  m  or  m, 

we  shall  have 

-fc  =  a*  (iM9  +  (i  +  f )  Mjn  +  (i  +  f )  Mpi*  +  •  •  •)  (23) 

§  13.    Numerical  v  alues  of  the  fundamental  quantities. 

Instead  of  effecting  the  preceding  transformations  analytically,  to  put  the  equa- 
tions (21)  into  numbers,  we  use  the  numerical  values  of  e,  y,  and  m  given  by 
Delaunay  in  his  Vol.  II,  pp.  801-802,  namely 

e  =  .054  8993  7  =  .044  8866  m  =  .074  8013  (24) 


1 6  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

We  then  find  from  his  expressions  on  p.  800 

a  =  0.996  4930  =  [9.998  474] a 
an  =  i.ooi  758072  =  [o.ooo  763] an 
a2n  =  0.998  245«2«  =  [9.999  237]«J« 
m  =  0.994  7437/2  =  [9.997  7n]?«  =  0.074  4082 
e  =  0.054  867  g  =  0.044  993 

We  also  find,  from  these  numbers,  the  following  values  of  the  required  partial 
derivatives  for  the  numerical  transformation 

^-  =  0.986  6910          s~  =  ~  °-°°7  37e  —  —  o-ooo  404     -j    =  +  0.006  857  =  o.ooo  308 

da  de  dg 

^  =  -0.0013750      ^=  +  0.99961  ^  =  +0.000202 

da  de  ,       dg 

^-  =  +0.0013530      5-  =  —  o.ooi  22e=  —0.000067      _j- =  + 1.002  324 

Then,  from  Delaunay,  II,  p.  236,  we  find 

L  =  i  .000  197 a2n  G  =  0.998  586a2n  //  =  0.994 

dL  dG  dH 

a       =  °tS°0  IS  a  a  ""  =  °<499    97  a"      = 


f^  T  r\  f~*  f)  J-T 

-^  =  —  o.ooo  o88a'n  -^-  =  —  0.052  4ioa2n  — —  =  —  0.052  185 a2n 

dL  dG  dH 

-v-  =—  0.0000073  n  --—  =—  0.0000353  n  ^-  =  —0.17947430 

§  14.  Formation  of  the  transformed  differential  equations. 
Let  us  now  return  to  the  equations  (19),  in  which  we  have  to  replace  c^  c2, 
and  c3  by  a,  e,  and  y.     We  have,  for  any  c, 

dc      dc  da      dc  de      dc  dg 
di  =  da  dt  +  de  dt  +  dgdt 
and 

dc       dc  da       dc  de      dc  dg 

• f         i      [      O 

da  ~  da  da       de  da      dg  da 
In  the  case  of  c^  we  have  from  (20) 

da  ~  da 


DEVELOPMENT  OF  THE  DIFFERENTIAL   EQUATIONS.  17 

so  that  the  numerical  expressions  need  not  be  repeated.     For  the  derivatives  of 
c2  and  c3  we  find 

a  ~-*  =  —  o.ooo  46iazn  ~  =  —  0.052  322a2n  ~  =  —  o.ooo  O28a2n 

da.  de  eg 

dc.  dc,  Be, 

a.fT-  =  —  0.002  ooi  a2n  -^  =  +  o.ooo  225  a2n  ~  =—  0.179  43°azn 

By  substitution  in  the  form  (22)  we  now  find 

dc  dc.  dc' 

-r  =  0.494  3°9«  M  ~=    =  —  o.ooo  777«'«  -~  =  o.ooo  &jifrn 

-•r-2  =  —  o.ooo  435«2«  ~B*  ~  ~  °-°52  2°9«2«  ~B~~  ~  °-oo°  O2Sa*n       (25) 

.,  3  =  —  0.002  033«2«  ~  =  o.ooo  igicfn  ^~3  =  —  0.179 

We  now  have  the  data  for  transforming  the  equations  (19),  p.  14,  so  as  to  express 
the  differential  variations  of  tf,  £,  and  y  instead  of  c^  c2,  and  c3,  and  to  express  those 
of  /0,  TTO,  and  00  in  terms  of  the  partial  derivatives  of  R  as  to  «,  £,  and  y.  For  this 
purpose  we  need  the  nine  partial  derivatives  of  a,  e,  and  y  as  to  clt  c2,  and  cs.  We 
shall  express  these  nine  derivatives  by  means  of  the  nine  numerical  factors 

a<»  e»1*  •••(*  =  1:2:3) 
defined  by  the  equations 

da  &?  ^7 

a.  =  az«  ^—  c  .  =  a  «  5—  7.  =  aw  3- 

dct  dcf  dct 

The  numerical  values  of  these  coefficients  are  most  expeditiously  found  in  the 
following  way.  Multiplying  the  first  three  equations  (19)  in  order  by  the  respective 

lactors 

da         da  da 

&;    ft;          ^ 

we  have 

da  dP.      da  dP.      da  dP. 

li_  \ 


with  similar  equations  in  Dt  e  and  Dty.    From  the  same  three  equations  we  have 

dCl  dc.  dc.  ,  dP 


da  de 

dc,  7  dc,  .  dc,  ,  dP 
..^  Dta  +  5-3  De  +  ~  Z>,7  =  -331 
da  ^  de  '  dy  "  dO 

It  follows  that  if  we  solve  these  three  equations  for  Dta,  Dte,  and  D,y  the  nine 
partial  derivatives  required  will  be  the  coefficients  of  the  second  members  in  the 


1  8  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

solution.  Replacing  the  coefficients  of  the  unknowns  by  their  numerical  values 
(25),  we  may  reduce  the  solution  to  that  of  three  numerical  equations 

0.494369^—  .  000777  Y+  •  00067  *Z  =  P 

—  .000435^—  .  05  2209  Y—  .  000025.?=  Q 

—  .002033^+  .  000191  Y—  .179538^=  ff 

The  solution  of  these  equations  so  as  to  express  JT,  Y,  and  Z  as  linear  functions 
of  P,  Q,  and  R  gives  the  following  values  of  the  factors  which  we  seek.  Along 
with  these  values  is  given  for  comparison  the  values  found  in  Action  of  Planets, 

Clffi 

p.  196,  where  the  numbers  are  the  coefficients  of  -  .     The  two  determinations 

JWj*»J 

are  completely  independent,  in  that  the  earlier  one  is  derived  from  the  analytic 
expressions  for  the  coordinates  of  the  Moon,  while  these  last  have  been  obtained 
from  Delaunay's  expressions  of  the  canonical  elements  L  G  H  in  terms  of  a,  e,  y. 

a,  =  -f  2.0228         Former  value  :     -f  2.0225 

a2  =  —  0.0301  —  0.0293 

«3  =  +  0.0075  +  0.0075 

el  =  —  0.0168  —  0.0169 

^=-19.1534  -ip^Si                                  (26) 

e3=  +  0.0026  +0.0017 

7,  =  —  0.0229  —0.0233 

72  =  —  O.O2OO  —  O.O2I6 

73  =-5-5700  -5-5704 

The  fundamental  differential  equations  for  the  variations  of  the  elements  now 

become 

BP.         dP.          dP. 

«,_-+  v_  +  o,^- 

*-st+t*-ti+t»it 

dP.          BP.          dPl 
- 


BPl 


(27) 


=  —  a  -3—  —  e.  -,-  —  7.  -^~ 
2  da         *  ce          2  cy 


-- 

3  da        3 

[Dim.  =  A/£-'] 


DEVELOPMENT  OF  THE  DIFFERENTIAL  EQUATIONS.  19 

In  order  that  we  may,  so  far  as  possible,  handle  only  pure  numbers,  with  speci- 
fications of  the  units  as  concrete  quantities,  we  shall  substitute  nt,  the  total  motion 
of  the  Moon  in  mean  longitude,  and  therefore  a  pure  number,  as  the  independent 
variable.  The  first  numbers  will  then  take  the  form  a2n2DHla,  etc. 

Since 

a'n2  =  - 
a 

the  equations  will  now  give 

'  <*> 


[Dim.  =o] 

with  five  others  formed  in  the  same  way  from  (27)  which  need  not  be  written. 

§  15.  Elimination  of  t  from  the  partial  derivatives  of  I,  TT,  and  6. 

An  important  remark  at  this  point  is  that  since  Pl  is  a  function  of  /,  TT,  and  6, 
the  three  quantities  a,  e,  and  y  enter  into  Pl  not  only  explicitly  but  implicitly 
through  n,  TT,  and  0,  so  that  the  complete  differential  variations  of  these  functions 
are 

dl      dL  dn  dtr      dir  dtr.  dO      d6n  dd. 

dt  =  df  +  "  +  'dt          *—3t+*>  +  <-3i         ar-rf  +  ».  +  'J 

Pl  being  a  function  of  the  six  quantities 

a,  e,  7,  /„  +  nt,  TT,  +  TT,*,  00  +  Qj 

its  complete  derivatives  as  to  a,  e,  and  y  are 


_          ,  , 

da  ~      da.      ^          dl  Ba  """  chr  da  "*"  d0   da 


with  similar  expressions  for  dP^Se  and  dP^dy.     Thus  we  have  for  any  canonical 
element  c 


dc       \  dc 


^to^a^dPidK 

dl  dc  +  dir  dc  "h  60  dc) 


20  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

The  complete  derivatives  of  /,  TT,  and  0  are  therefore 

.t(dn      BP^dn      BP,  d-^      dP^d 
\dt~'  dl  dCl      dw  dCl  "  d0  dCl 


dt  ~  dt 

dir      dir  (  cfrr        dP  dn       dP  STT        BP  80 


V    d_P^"    ^3^    a^e^\ 

"   dl  ~6c3      S-n-  dc3  "  d0  dcj 


dt  -  dt         '  v  dt 

It  is  a  fundamental  theorem  of  the  development  of  the  planetary  coordinates  in 
periodic  series  that  the  terms  of  these  equations  containing  t  as  a  factor  all  vanish.* 
The  values  of  /,  TT,  and  6  are  therefore 

*  A  demonstration  of  this  theorem  in  the  most  general  case  is  found  in  the  author's  paper  On  the  General  Inte- 
grals of  Planetary  Motion  :  Smithsonian  Contributions  to  Knowledge,  1874. 


CHAPTER  III. 

DEFINITIVE  FORM  OF  THE  DIFFERENTIAL  VARIATIONS 
OF  THE  ELEMENTS. 

§  16.  The  differential  equations  (27)  in  the  form  (28)  are  the  fundamental  ones 
of  our  problem,  the  integration  of  which  is  to  be  effected.  This  need  be  done  only 
to  terms  of  the  first  order  as  to  the  disturbing  function.  This  amounts  to  saying  that 
we  regard  the  second  members  of  the  equation  as  known  functions  of  the  time,  and 
that  the  required  integration  is  to  be  performed  by  simple  quadrature. 

We  begin  by  studying  the  general  form  of  the  function  f\.  Besides  A^?,  this 
function  consists  of  two  parts,  one,  7?,  arising  from  the  direct  action  shown  in  §  7, 
and  the  other  flp  arising  from  the  indirect  action.  We  have  reduced  both  these 
parts  to  the  general  form 

Ax2  +  By*  +  Cz*  +  2Dxy  +  lExz  + 

The  coefficients,  A,  J5,  etc.,  are  functions  of  the  heliocentric  coordinates  of  two 
points:  the  centre  of  gravity  G  of  the  Earth  and  Moon,  and  that  of  the  planet. 
They  are,  therefore,  regarded  as  independent  of  the  elements  of  the  Moon's  orbit. 
The  variables  x2,  y2,  etc.,  being  functions  of  the  geocentric  coordinates  of  the  Moon, 
are  independent  of  the  position  of  the  planet,  and  contain,  besides  the  six  lunar  ele- 
ments proper,  the  major  axis  and  eccentricity  of  the  Earth's  orbit  around  the  Sun. 
The  arguments  on  which  the  coefficients  A,  B,  etc.,  depend  are  g±  and  g'  .  The 
coordinates  #2,  y2,  etc.,  depend  on  the  four  arguments  /,  IT,  0,  and  g'  .  It  follows 
that  the  terms  of  Pl  depend  on  the  five  arguments 


Although  the  two  actions,  the  direct  and  indirect,  admit  of  being  treated  together 
by  combining  the  corresponding  coefficients  of  #2,  y2,  etc.,  yet  the  coefficients  are 
so  different  in  their  form  and  origin  that  it  will  be  better  to  treat  them  separately. 

§  17.    Reduction  of  the  equations  for  the  direct  action. 

We  begin  with  the  development  of  ^?,  as  given  by  (13)  and  (14).     Since 

x1,  y1,  etc.,  each  =  a  pure  number  x  a2 
A,  B,  C,  etc.,  each  =  a  pure  number  -=-  a' 


21 


22  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

it  follows  that  R  may  be  developed  in  the  form 

R-fr^H  (30) 

H  being  a  pure  number. 

When  the  fundamental  equations  are  taken  in  the  form  (28),  and  /*,  is  replaced 
by  R  expressed  in  terms  of  A,  the  second  members  will  all  take  the  common 

constant  numerical  factor 

37«<  «» 
2  p  a>* 

This  factor  may  be  simplified  by  the  fundamental  relations 

«V  as  ft  a'  V2  =  m'  +  p 

where  p.  and  m'  are  the  respective  masses  of  Earth  -j-  Moon  and  of  the  Sun. 

Owing  to  the  minuteness  of  fi  relative  to  m'  (i  1330000  -(-)  we  may  drop  it  from 
the  quotient,  thus  obtaining 

m'  <?      nft        2 

i  =  "-  •   ^=  7W 

M   a'3       «* 
The  factor  thus  reduces  to  the  pure  number 

*  ^  nf 
2  me 

The  ratio  m^\  m'  is  what  is  commonly  taken  as  the  numerical  expression  of  the 
mass  of  the  planet.     We  shall  write 

M=  ?  ->'  =  0.008  392  86  ~4 

2  »«'  »2r 

The  numerical  values  of  M  for  the  four  planets  whose  action  is  to  be  determined 
are  as  follows: 

•' 

^  M 

Venus  408  ooo  o".oo4  242 

Mars  3  093  500  o  .000  560 

Jupiter  1047.35  i   .653 

Saturn  35OO  o  .4947 

We  have  next  to  consider  H  and  its  derivatives.     As  this  quantity  has  been 
above  introduced  we  have 


(31) 


The  terms  in  E  and  fare  omitted  here,  owing  to  their  minuteness. 


DIFFERENTIAL  VARIATIONS  OF   ELEMENTS.  23 


We  have  now  to  deal  with  two  sets  of  factors : 

1.  The  planetary  fac 

2.  The  lunar  factors 


i.  The  planetary  factors,  a'3 A,  a'3!?,  etc. 


x2   y"1  z1    2xy 
~tf'  ~a"~tf'  ~~Jr>  CtC< 
for  which  we  use 

f2,  if,  £2,  2l~r),  respectively 

§  1  8.  Notation  of  the  Planetary  Factors.  The  development  of  these  requires 
numerical  processes  which,  owing  to  their  length  and  their  distinctive  character, 
are  given  in  Part  II.  We  shall  therefore  assume  this  development  to  be  effected, 
referring  to  Part  II  for  the  methods  and  numerical  results.  Considering  the  latter 
in  their  general  form,  we  remark  that  these  coefficients  being  of  dimensions  Z~8, 
if  we  compute  their  values,  taking  the  Earth's  mean  distance  as  unity,  the  numbers 
obtained  for  the  several  coefficients  A^B,  etc.,  will  readily  be  the  values  of  a'3A, 
a'3B,  etc.  We  shall  therefore  put 

a'"  A  =  2  (A.  cos  JV4  +  A,  sin  JVt) 

a'3S  =  2  (£c  cos  7V4  +  B.  sin  7V4)  (33) 

a'*C  =  2  (Ct  cos  7V4  +  C.  sin 

•  •••• 

where  each  argument  is  of  the  general  form 


/4  being  the  mean  longitude  of  the  planet,  measured  from  a  point  which  we  shall 
take  as  that  corresponding  to  the  earth's  perihelion. 

§  19.  Notation  of  the  lunar  factors.  We  have  shown  in  Action,  Chapter  II, 
how,  from  Delaunay's  results,  the  squares  and  products  of  the  Moon's  coordinates 
may  be  developed  in  the  general  form 


r  (34) 

2%  t]  =  2*4  sin  N  2%%  ==  2*s  sin  N  277?  =  2*6  cos  N 

Here  the  K  are  functions  of  «,  e,  y,  a',  and  e',  and  the    arguments  JV  may  be 
expressed  in  the  general  form 

N=  il  +I'TT  +  i" 


These  developments  comprise  all  the  quantities  necessary  to  the  formation  of 
and  its  derivatives. 


24  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

§  20.    Numerical  form  of  the  fundamental  coefficients.     The  condition 

A  +  B  +  C  =  o 

enables  us  to  reduce  by  one  the  number  of  terms  in  //,  and  at  the  same  time  to 
simplify  the  computation.     We  have  the  identity 


A?  +  B,f 
Replacing  A  -\-  B  by  —  C  there  results 

A?  +  Br?  + 

Putting,  for  brevity, 

K=  \a'\A  -  B)  C;  =  0'3C  p»  =  .£J  =  p  +  i,»+(? 

which  will  make  K,  C»  and  Dl  pure  numbers,  we  shall  have 


The  planetaiy  factors,  A,  C^  and  D^  are  taken  as  developed  in  a  double  trigo- 
nometric series  from  the  equations  (33),  by  putting 


We  shall  then  have  for  //the  double  trigonometric  series 
//=  2  (Kc  cos  7V4  +  Kt  sin  7V4)  (*,  -  «,)  cos  7V^ 

-^(\CC  cos  Ar4  +  ^  C.  sin  /VJ  («,  +  *2  -  2«s)  cos  .V  (35) 

+  2  (Z>c  cos  7V4  +  /?,  sin  7V4)*4  sin  N 
Introducing,  for  brevity, 

-/*  =  M*!  -  *2)  ?=aJ(«l+*l)-"s  (36) 

the  terms  of  the  lunar  factors  will  be  expressed  by 

(f2  —  7/2)  =  2/>  cos  7V  p1  —  3?2  =  2^  cos  N  ifr  =  Kt  sin  N. 

Every  combination  of  a  planetary  argument  Nt  with  a  lunar  argument  yV^  will  give 
rise  to  a  set  of  terms  in  H  of  the  form 

H=  h.  cos  (JV+  .A7,)  +  A.  sin  (AT+  /VJ  +  ///  cos  (TV-  7V4)  +  ///  sin  (IV-  Nt)       (37) 

where 

^c  =  K.I  ~  \  C,g  -  JZ?  *4  ^  =  -  KJ  -  J  Cf?  +  JZ?.*4 

(38) 
//.  =  K.p-\  C,g  +  \Df,  h>  =  -JTtp  +  i  C.g 


DIFFERENTIAL  VARIATIONS  OF  ELEMENTS.  25 

The  partial  derivatives  of  If  as  to  a,  e,  and  y  are  to  be  found  from 

Dhc  =  KcDp  -  J  CcDq  - 


de 

a% 

dy 

with  three  other  sets  formed  by  replacing  p,  q,  and  /c4  in  (38)  by  their  partial  deriva- 
tives. These  derivatives  of  Ac,  hn  Ac',  and  hi  being  substituted  in  (37)  give  the 
required  partial  derivatives  of  H.  In  forming  the  derivatives  as  to  /,  TT,  and  6  we 
note  that  these  quantities  enter  only  through  the  arguments  jV,  in  which  they  have 
the  respective  coefficients 

Their  formation  is  therefore  a  simple  algebraic  process  after  H  is  developed. 

The  elements  e  and  y  also  enter  R  only  through  H.    But  a  appears  both  in  //", 
which  is  a  function  of  m,  and  in  the  factor  a2/ a'3.     We  therefore  have  from  (30) 

dR  a1  /  dH 


For  consistency  in  form  and  notation  we  shall  put 

D'H=  2H+  ™  (40) 

It  may  be  remarked  that  the  formation  of  D' H  may  be  effected  by  the  general 
operation  indicated  in  (23),  by  supposing  H  developed  in  powers  of  m  and  putting 

M=d>H 

so  that 

/=  2 
We  then  have 

and 

dH 


The  sum  of  this  -\-^H  gives  D'  H  as  above  expressed.     In  forming  this  sum  we 
need  not  use  the  analytic  development  of  2//,  which  is  necessary  to  form  dH/da, 
but  may  use  the  numerical  development  when  it  is  more  accurate. 
The  partial  derivatives  of  R  as  to  a,  e,  and  y  are 

a2  dR  a2  dH  dR  a1  dH 

3  ij»    __    T)t  If 

-    9  fflA         '<  •*-*    J-* 


__  _  _ 

1         -    9  fflA         '<  •*-*    J-*  ~-^         -   iS  Irl  A          •   ~~^  ~^          —   tffff,         *   ~^ 

Da       2    ta>i  de       2    4fl'3  de  dy       «    4a's  dy 


26  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

§  21.  The  fundamental  equations  in  the  form  (28)  for  the  direct  action  now 

become 

dH        dH        djf\ 

dH        dH        dH\ 

i  ^  ~^i    ~T~  ^«~a         i"  ^a    a  ZT    I 

ol  OTT  uu  f 

dH\ 


-DJ,  = 

+e2d+y2  (42) 


§  22.  We  have  next  to  show  how  the  second  members  of  these  equations  may  be 
most  readily  reduced  to  numbers.  There  being  a  certain  number  of  lunar  argu- 
ments N  and  also  a  certain  number  of  planetary  arguments  N4,  it  will  conduce  to 
simplicity  to  carry  forward  the  quantities  depending  on  the  argument  of  each  class 
as  far  as  possible  before  making  the  combination. 

Each  lunar  argument  being  of  the  general  form 


and  each  planetary  one  of  the  form 

Nt  =  k'g>  +  klt 

it  follows  that  by  putting  G  for  the  general  value  of  the  combined  final  argument, 

N±N« 

G=il+  i'-n  +  i"9  +  (j±  k')g'  ±  k?t 

the  general  iorm  (37)  of  H  may  be  written 

osG  +  /iisin  G) 


The  derivatives  of  H  as  to  7,  IT,  and  6  are 

-^j-  =  2  (—  t7ic  sin  G  -f  ihs  cos  G)  -=—  =  S(—  i'he  sin  G  +  i'ht  cos  G) 


-QQ  =  2(-  f'ht  sin  G  +  i"ht  cos  G) 

Substituting  these  values  in  (41)  and  putting 

a  =  /a,  +  t'a2  +  i"a^  e  =  iel  +  i'e^  +  i"et  g  =  f^  +  t'j2  +  S"y3  (43) 


DIFFERENTIAL  VARIATIONS  OF   ELEMENTS.  27 

the  equations  (41)  become 

Dp.  =  M(a.ht  cos  G  —  a/ic  sin  G) 

Dnte  =  M(eht  cos  G  —  eAc  sin  G)  (44) 

Z?n(7  =  M(ght  cos  G  —  ghc  sin  G) 

Every  combination  of  a  lunar  argument  TV  with  a  planetary  argument  N4  gives 
rise  in  each  derivative  of  an  element  to  four  terms,  which  we  shall  express  in  the 
form 


DMa  =  /&„,  .  cos  (IV  +  N.)  +  *„  .  sin  (JV+  7V4)  +  A,t  .'  cos  (2V-  N.)  +  *..  /  sin  (2V-  Nj    (45) 
Replacing  ^,  and  //,.  in  (44)  by  their  values  (38)  we  have  for  each  combination 


(46) 


(47) 


-  \MC,gq 

+  \MD&*< 

(48) 


§  23.  We  now  reduce  in  a  similar  way  the  group  (42).     We  have  for  each 
argument, 


D'H  =  D'hc  cos  G  +  D'h,  sinG  ene  +      r*6  =      f  COS  C  +       sin  6^ 

CC         oe  5g  ^77        C7  Cy 

Replacing  hc  and  ^g  by  their  values   (38)  and  substituting  the   resulting  partial 
derivatives  in  (42)  we  have  results  which  we  may  write  in  the  form 


-  Z>,,/0  =  //,,  .  cos  (N+  7V4)  +  A,,,  sin  (N+  7V4)  +  //,,/  cos  (IV-  Nt)  +  hlt.'  sin  (IV-  2Vj 
-Z>n(7r0=  h,ie  cos  (N+  N,}  +  h,it  sin  (^V+  yV4)  +  h,t  .'  cos  (IV-  N,}  +  A,t  ,'  sin  (IV-  JVt)  (49) 
-1)JS9  =  //„,  e  cos  (7T+  A1,)  +  //,,  .  sin  (yY  +  JVt)  +  hti  /  cos  (JV-  vV4)  +  A..  /  sin  (N-  JVt) 


28  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

where  the  values  of  the  coefficients  are  found  by  the  following  computation.     For 
each  lunar  argument  we  form 


(50) 


Then  for  each  pair  of  arguments 

//,,  .  =  MKCL'  -  \MCCL"  -  \MD.LI  hlt  /  =  +  MKCL>  -  \MCCL" 

hlt  ,  =  MK,L'  -  \MC,L"  +  \MDCL,  hlt  /  =  -  ^ff;Z'  +  \MCtL" 

hfi  .  =  J/^P'  -  \MCJP"  -  \MD.P,  h,t  /  =  +  J/7T/"  -  \MCCP" 

h,t,  =  MKtP'  -  \MC,P"  +  \MDCP,  AWi.'  =  -  MKf  +  \MC.P"  +  \MDJ\ 

htt  ,  =  MKCR'  -  \MCR"  -  \MDtRi  htt  c'  =  +  MKCR'  -  \MCCR" 

hlt  ,  =  MK,R>  -  \MC,R"  +  \MDCR,  /<„,  ,'  =  -  MKtR  '  +  \MC.R" 


§  24.  Development  of  the  indirect  action, 

The  fundamental  equations  for  the  indirect  action  are  found  from  (28)  by  replac- 
ing PI  by  the  function  flp  defined  in  (12).  We  first  replace  the  coefficients  SZ'by 
the  following: 

v  etc. 


Taking,  as  we  do  throughout  this  work,  the  mean  Sun  as  the  origin  of  longi- 
tudes, the  true  longitude,  v',  will  be  replaced  by  the  Sun's  equation  of  the  centre 

=  E.     We  also  put 

r' 


With  these  substitutions  the  equations  (n)  will  be  replaced  by  others  which  may 
be  written  thus:     Put 


G  =  f  7-j-s  sin  zElv'  +  %r-3  cos 

«/-  f^rV  (52) 

/=  |ri-3  cos  2ESv'-$r-s  sin 


Then 

A'  =  -G-J  B'  =  G-J  C'  =  2j  D'  =  I 

E'  =  f^-3  cos  E  sin  /8'  F'  =  \r~*  sin  E  sin  £' 


DIFFERENTIAL  VARIATIONS  OF   ELEMENTS.  29 

These  substitutions  lead  to  the  replacement  of  expression  (12)  by 

n,-=£*'  (54) 

where 

H'  =A'?  +  B>'n*+C>?+2D'l;r,+  -.-  (55) 

This  function  //',  a  pure  number  in  dimensions,  will  hereafter  be  used  as  a  fun- 
damental quantity  instead  of  flp. 

By  replacing  P,  by  this  value  of  flp  in  (27)  the  second  members  in  the  form  (28) 
take  the  common  factor 


Ha'3 


and  the  differential  variations  of  the  elements  become 

dH'          dH'          dH' 


dH'          BH 
Dnte  = 


BH'\ 

-^  J  (56) 


dH'          dH'          dH' 


(57) 


We  have  next  to  develop  the  values  (52)  of  G,  J,  and  /  in  terms  of  the  mean 
anomaly  £•'.  This  may  be  done  by  means  of  Cayley's  tables  in  the  Memoirs  of 
the  Royal  Astronomical  Society,  Vol.  XXIX,  or  the  development  given  by 
Leverrier  in  Annales  de  V  Obseivatoire  de  Paris,  Vol.  I.  Dropping  unnecessary 
terms  and  powers  of  e'  we  have 

r~*  cos  lE  =  i  —  |e'2  +  (3^'  —  tye'*)  cosg'  +  Ij-e'1  cos  2g' 
r~3  sin  2E=  ($er  -  *£e'*)  sing-'  +  -^V'sin  2g' 

r~*  =  i+  |  e'2  +  (3e'  +  Qe'3}  cos  g'  +  \e<*  cos  2g  (58) 

r~*  cos  E  =  i  +  $e'  cos  g' 

r~*  sin  E=  "i 


30  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

The  expressions  for  G,j,  and  /thus  become 

G  =  {(6e'  -  -9/V3)  sing-'  +  -\V2  sin  2g'}&v' 

+   {9  __  45.e/2  +  (]£gt  -  -VgV3)  COS£"'  +  lf^'2  COS  2g'}Sp' 

/=-(!  +  &'*  +(!«'  +  IX3)  cosg-'  +  -\V2  cos  2£-'}V  (59) 

/=  {|  -  -¥-e'2  +  (K  -  We/s)  cos£"'  +  ¥e/2  cos  2£-'}&>' 

-  {(9*'  -  -VgV)  sing-'  +  ifV2  sin  ig'}  &p' 


In  reducing  these  expressions  to  numbers  I  take,  with  Delaunay  and  Brown,  the 
value  of  e'  for  1850 

e'  =  .016  771 

With  this  datum  the  expressions  for  G,J,  etc.,  become 
G  =  (0.10058  sing-'  +  0.00359  sin  2g')8v' 

+  (2.24842  +  O.II3I3  COS£-'  +  O.OO538  COS  2g')Spr 

J=  (0.75032  +  0.03775  cos  g1  +  0.00095  cos  2g-')Sp'  (60) 

7=  (1.49895  +  0.07542  cos^-'  +  0.00359  cos  2g')Sv' 
—  (0.15087  sin£"'  +  0.00538  sin  2g-')$p' 


§  25.  Abbreviated  coefficients  for  the  indirect  action.     Since 

A'  +  B'  +  C>  =  o 
we  have,  as  in  the  direct  action, 

H>  =  \(A'-  B')(?  -  rf)  -  \C>  ( 

Replacing  A',  £',  and  C'  by  their  values  (53) 

H>  =  -  G?  -  J  - 


As  the  last  two  terms  of  H'  are  important  only  in  some  exceptional  cases,  we 
postpone  their  development  to  Part  IV. 

With  the  notation  of  (36),  we  have  for  each  lunar  argument 

H'  =  (-  2  Gp  cos  N—  2jq  cos  N+  7*4  sin  N)  (61) 

The  planetary  factors,  G,J,  and  /are  to  be  developed  in  a  periodic  series  of  the 
same  form  as  that  for  A,  B,  and  C,  so  that,  for  each  planetary  argument  N±  we 
shall  have 

G  =  Gc  cos  JVt  +  G,  sin  7V4        /  =  Jt  cos  JVt  +  J.  sin  7V4        /  =  /.  cos  7V4  +  /.  sin  7V4    (62) 


DIFFERENTIAL  VARIATIONS  OF   ELEMENTS.  31 

With  these  values  we  shall  have  H'  developed  in  a  double  series   in  which  for 
each  pair  of  arguments  j^Vand  JV4,  H'  will  have  the  four  terms 


H'  =  ht  cos  (N+  JV4)  +  hc'  cos  (JV-  Nt}  +  ht  sin  (N  '+  7V4)  +  h,'  sin  (IV-  vVJ      (63) 
where 

¥.«i  V  =  ~Gep-  Jcq  +  K.«< 


h.  =  -G.p-  J.q  +  J/A  A/  =  G,p  +  Jtq  +  K*4 

Expressing  the  differential  variations  of  the  elements  in  the  same  form  as  before 
we  shall  find 


Ka/0        hJ  =  ™\  G.zp 

(64) 


with  two  other  sets  of  equations  found  by  replacing  a  and  a  by  e  and  e  for  the  set 
in  e,  and  by  y  and  g  for  the  set  in  y.     Also, 


ht>.  =  nf(-  GeL'  -JCL"  -  J/.A)         h^'  =  m\-  GCL'  -  JCL" 

(65) 
Alt.  =  m*(-  G.L'  -J.L"  +  KA)        *«,.'  =  m2(G,L'  +  J,L"  +  \IcLt) 


with  two  other  sets  formed  by  replacing  /  and  L  by  TT  and  P,  for  the  set  in  IT,  and 
by  6  and  R  for  the  set  in  0. 

Comparing  these  with  the  corresponding  coefficients  (51)  for  the  direct  action 
we  see  that  the  equations  for  the  indirect  action  may  be  formed  from  those  of  the 
direct  action  by  replacing 

K,  \C  and  D  by  —  G,  J,  and  /;  and  also  Mby  m? 

It  also  follows  that  the  two  actions  may  be  combined  by  replacing  in  the  expressions 
tor  the  coefficients  h,  given  in  (46),  (47),  (48)  and  (51), 

MK  by  MK-  m2G  ;  \MC  by  \MC  +  n?J  ;  MD  by  MD  +  ntl    (66) 

We  shall  make  this  combination  to  save  labor  in  the  formation  of  the  products,  but 
shall  give  the  separate  parts  of  the  coefficients,  so  that  the  parts  of  each  term  due 
to  the  respective  actions  may  be  readily  found. 

§  26.  Integration  of  the  equations.  The  integration  is  effected  by  multiplying 
each  coefficient  by  the  quotient  of  the  mean  motion  of  the  Moon  by  the  motion  of 
the  argument  itself,  which  factor  is 

n  .  ,  , 

"  =  in  +  t'lr,  +  i"0l  ±  (J  +  k')n'  ±  kn4  ^  7' 


32  ACTION   OF  THE  PLANETS  ON  THE  MOON. 

The  reciprocal  of  this  factor,  which  we  may  use  as  a  divisor,  is 


a  form  most  convenient  for  numerical  computation. 

We  shall  thus  have  for  the  perturbations  of  the  elements  corresponding  to  each 
pair  of  lunar  and  planetary  arguments 


Sa  =  vh^e  sin  (IV  ±  N^  )  -  vh^t  cos 

Se  =  vh,iC  sin  (N±  JVt  )  -  vhtt.  cos  (IV  ±  JV4  )  (68) 

S7  =  vhy<e  sin  (N±  JV4  )  -  vAy>.  cos  (JV±  JV4  ) 

/0  =  -  «/>&,_„  sin  (JV  ±  JVt  )  +  vhti,  cos  (7V^±  7V4  ) 

r0  =  -  I/A,,,  sin  (1V±  JVt  )  +  ^ffi.  cos  (IV  ±  7V4  )  (69) 

^0  =  -  i/A,ie  sin  (N±  7V4  )  +  f//9,,  cos 


Practically  we  use  the  perturbation  of  n,  the  mean  motion,  instead  of  a.     From 
the  relation  of  §  12,  {$,  we  have 

Dfi  =  —  \nDp. 

Thus  the  first  equation  (68)  is  replaced  by 

Sn  =  -  \vnh^e  sin  (IV  ±  7VJ  +  f  wiA.,.  cos  (^±  7V4)  (70) 


§  27.  We  pass  next  to  the  inequalities  of  the  actual  mean  longitude,  /,  and  of  the 
perigee  and  node,  IT  and  6.     Taking  the  equations  (29)  for  these  quantities 


TT  =  TTO  +       jrf/  0  =  0,, 

the  complete  expressions  are 

8/  =  S/0  +  fSndt  Btr  =  57r0  +  fSv^t  80  =  800  +  fse^t  (71) 


The  motions  n,  TT^  and  ^  are  functions  of  the  elements  a  (or  a),  e,  and  y.  n  is 
given  by  the  relation  a3«2  =  /x,  while  TTJ  and  ^  have  been  developed  by  Delaunay, 
whose  results  are  found  in  Comptes  Rendus,  Vol.  LXXIV,  1872,  I,  and  are  repro- 
duced in  part  in  Action,  p.  190. 

*.-£*•+£«•+£»*     »>-%»•+%><+%*> 

From  (70)  and  (71)  we  thus  have,  in  the  variation  of/,  the  terms 

0J*  =  -  !  <  .  sin  (^v±  W  +  1"*.,  .  cos  (^±  ^*)  (7  2) 


DIFFERENTIAL  VARIATIONS  OF   ELEMENTS.  33 

arising  from  the  variation  of  n.     Integrating  and  including  the  value   of  8/0  we 
shall  have  for  the  complete  perturbation  of  the  mean  longitude 


U  =  lt  cos  (N±  JVt)  +  I,  sin 
where 

lc  =  IMai  c  +  Vhlt  ,  =  v  (f<  .  +  //,,  .)  /.  =  f  itt.,  .  -  vhlt  ,  =  v  (f<  .  -  A,,  .)        (73) 

From  the  Delaunay  developments  in  powers  of  m  are  found 

°433  1« 

(74) 


BIT.  BIT  dir. 

-      =  -  .01480  -+  =  -  .ooio42«  =  -  -°°433  1« 


B0.  B0.  80l 

-^  =  +  .00377  j*  --  .ooi292«  &j  =  +  - 

Substituting  these  values  and  the  values  (68)  and  (70)  we  find  that  by  putting 

7T,  „  =  .02220/Ja  „  —  .00104/&e  „  —  .00433^,  e      7T1|4  =  .02220/1,,  —  .OOIO^,  —  .00433^,. 


(75) 
t  ,—  .00129^,  +  . 


we  shall  have 

STT,  =  v«  {irlt  e  sin  (vV^i  TV7;)  —  ir^  .  cos 


l  =  I/M  {,,  c  sn        ±     ,  -    lt  ,  cos 
Then  by  integrating  we  have  the  terms 

STT  =  -  j/V,  „  cos  (TV7"*  ^V4)  -  i/V,  .  sin 


(76) 
cos       i     4  -  1/,,  .  sn 


Adding  the  values  (69)  we  have  the  complete  periodic  perturbations  of  IT  and  0 
expressed  in  the  form 

STT  =  TTC  cos  (JV±  .VJ  +  w,  sin  (TV^i  7VJ  8^  =  0o  cos  (^Vd=  JVJ  +  0,  sin  (JV±  JVJ 

where 

•"•c  =  "^-r,  .  -  "X,  e  =  K7^,  .  -  ^1,  •)  7T.  =   -  !///,_  „  -  Z^TT,,  ,  =   -  !</*„,  „  +  OTT,,  ,) 

(77) 


§  28.    Treatment  of  the  non-periodic  terms  in  R. 

In  the  preceding  integration  we  have  supposed  all  the  arguments  to  be  of  the 
form  GQ  +  N/.  We  have  now  to  consider  the  special  case  in  which  N  vanishes. 
In  the  case  of  the  direct  action  this  occurs  when,  in  the  pair  of  arguments  which 

form  G, 

i=i'  =  i'  =  k  =  o  j  ±  k'  =  o 

We  shall  then  have  in  H  a  term,  H=  thn  which  we  shall  call  hc  simply.     It  will 
be  affected  by  a  minute  secular  variation  which  we  need  not  consider  at  present, 


34  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

In  the  case  of  the  indirect  action  we  note  that  the  coefficients  of  Sz/,  Sp',  and  S/3', 
as  found  in  (59),  are  developed  in  the  general  form  2/£,  sin  ig'  or  kt  cos  ig',  in 
which  /&0,  being  a  function  of  the  eccentricity  of  the  Earth's  orbit,  is  a  function 
of  the  time.  The  coefficients  of  all  the  terms  arising  from  the  indirect  action  are 
therefore  aftected  by  a  secular  variation. 

The  perturbations  8v  '  and  8p'  contain  terms  independent  of  the  mean  longitude 
of  the  disturbing  planet,  which  may  be  treated  separately,  namely: 

(1)  A  constant  term  in  8p'. 

(2)  Terms  of  the  form  c  ^  ig'  in  8v'  and  S/>'. 

(3)  The  secular  variation  of  e'  and  of  p'  . 

Omitting  for  the  present  the  powers  of  t  above  the  first,  we  shall  have  in  8t>  '  and 

8p'  terms  of  the  general  form 

(c  +  c'f)  £  ig' 

The  product  of  these  into  (59)  gives  rise  to  terms  of  G,  J,  and  /of  the  same  form. 
When  we  form  the  products  of  these  terms  by  f  2,  rf,  etc.,  we  shall  have  in  H'  terms 

of  the  form 

h  +  h'nt  +;  •  • 

Substituting  the  derivatives  of  the  non-periodic  direct  term  in  (41)  and  (42),  and  of 
the  indirect  term  in  (56)  and  (57),  omitting  terms  in  /,  and  putting  for  brevity 

P%  =  Mhc  +  nth 
we  find 


-  Djr.  - 


(78) 


-         .  ,       3  y3        .  A" 

Adding  in  the  terms  multiplied  by  «/,  these  three  equations  may  be  written 

DJ,  -  -  A,  -  h,Ht  Dnr,  =  -  h>  -  h'nt  DJ0  =  -  V  -  A^'nt       (78') 

The  integration  of  (78)  and  (78')  will  give 

?>a  =  V  ;  S/o  =  V0  -  V  -  J  V2'2 

Se  =  V  ;  &r.  =  V.  ~  >>»'»*  ~  P»'«V  (78") 

By  =  8o7  ;  S00  =  Bad0  -  h0'nt  -  J/;/V/< 

80  designating,  in  each  case,  the  arbitrary  constant  of  integration. 


DIFFERENTIAL  VARIATIONS  OF   ELEMENTS.  35 

The  completed  expressions  ior  /,  rr,  and  0  are  to  be  found  by  the  equations 

d«  *  &*,  «        dTr,  .         dir.  ,  60.  .         80.  .         60.  . 

Sn  =  i-  &a  STT.  =  ^-'  8a  +  -,  .-'  &e  +  -=-1  87  80.  =  -^  Sa  +  -^  Se  +  -^  87    (79) 

da.  da  de  dy  da  de  tty 

87  =  8/0  +  f  $>ndt  Sir  =  STTO  +  /  Str^t  $0  =  S<?0  +  f  W^t  (80) 

In  these  equations  the  perturbations  (78")  are  to  be  substituted.  In  doing  this 
the  arbitrary  constants  8/0,  807r0,  and  80#0,  being  merely  constant  corrections  to  /,  TT, 
and  0,  may  be  dropped  as  unimportant  to  the  theory.  We  shall  then  have  from 
(78")  and  (80) 


'  V  +    1  V  +     *o7  -  * 

§  29.  Adjustment  of  the  arbitrary  constants.  Values  are  next  to  be  assigned  to 
the  arbitrary  constants  80a,  S0g,  and  80y.  We  shall  do  this  so  as  to  satisfy  the  condi- 
tions that  the  coefficient  of/  in  S/,  of  sin  g  in  the  mean  longitude,  and  of  sin  (1—6} 
in  the  latitude,  shall  all  remain  unchanged.  The  first  of  these  conditions  gives 

-  f  «  V  =  V  or  V  =  -  ¥l»  (82) 

We  thus  have 


The  determination  of  80e  and  80y  must  await  the  computation  of  the  periodic 
terms  depending  on  the  arguments^  and  1—0,  which  is  found  in  Part  IV.  The 
increments  in  the  motions  of  TT  and  0  now  become 


(83) 


§  30.  Opposite  secular  effects  of  the  direct  and  indirect  action  of  a  planet 
near  the  Sun. 

An  important  theorem  of  the  planetary  action  on  the  Moon  is  that  as  the  planet 
is  nearer  the  Sun,  not  only  does  each  form  of  action  become  smaller,  but  the  two 
forms  tend  to  cancel  each  other,  so  that  when  the  mass  of  the  planet  can  be  con- 
sidered as  simply  added  to  that  of  the  Sun,  the  non-parallactic  perturbations  vanish. 


36  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

To  find  the  effect  of  the  direct  action  in  this  case,  let  the  values  of  x4,  j4,  and  24 
in  (14)  be  so  small  that  they  may  be  neglected  in  comparison  with  x',  y',  and  z'. 
Then  A  will  merge  into  r'  and  we  shall  have 


For  the  indirect  action  we  remark  that  the  only  effect  of  the  action  of  the  planet 
on  the  position  of  the  Earth,  after  so  adjusting  the  constants  of  integration  that  the 
mean  motion  shall  remain  unaltered,  is  to  increase  the  mean  distance,  so  that 
instead  of 


.3     .2  . 

a'  n'  =  m' 


we  shall  have 

This  gives,  for  the  perturbation  of  a' 


.3     ,2  . 

a'  n'   =  m   -f-  »/4 


and  the  eccentricity  e'  being  unaltered 


- 

3m> 


The  corresponding  part  of  flp  is  found  from  (7)  by  assigning  the  increment  r'Bp' 
to  r'.     We  thus  have 


«,-- 


This  cancels  the  value  of  R  found  above. 


PART  II. 

NUMERICAL  DEVELOPMENT  OF  THE   PLANETARY 

COEFFICIENTS. 


40  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

that  we  could  not  be  sure  of  this  point  without  actual  computation.  In  the  case 
of  the  Hansenian  inequality  of  long  period  due  to  the  action  of  Venus  it  was  shown 
that  the  perturbations  in  question,  considered  individually,  were  nearly  of  the 
same  order  of  magnitude  as  the  coefficients  to  be  determined.  This  proceeded 
from  the  fact  that,  even  when  we  consider  only  the  formulae  of  the  elliptic  motion, 
the  coefficients  of  the  term  in  question  are  in  the  nature  of  minute  residual  differences 
of  large  quantities.  In  view  of  the  undoubted  fact  of  some  apparent  inequalities 
of  long  period  in  the  motion  of  the  Moon  of  which  theory  has  yet  given  no  expla- 
nation, it  seems  necessary  to  exhaustively  discuss  every  possible  mode  of  action 
which  might  affect  the  result. 

The  most  effective  and  certain  way  which  the  author  could  devise  to  over- 
come this  difficulty  was  to  employ  the  purely  numerical  development  sometimes 
called  "mechanical  quadratures,"  but,  more  exactly,  that  of  induction  of  general 
formulas  from  special  values.  It  is  true  that  the  numerical  computations  required 
by  this  method  would  be  very  voluminous,  possibly  more  so  than  those  by  other 
methods.  But  the  use  of  the  method  has  the  great  advantage  that  the  computations 
are  made  on  a  simple  and  uniform  plan,  which  can  be  executed  by  routine  com- 
puters, and  in  which  the  complexity  incident  to  the  analytic  treatment  does  not  enter 
at  all.  Another  important  advantage  of  this  purely  numerical  method  is  that  the 
mutual  periodic  perturbations  of  Venus  and  the  Earth  can  be  taken  account  of  from 
the  beginning.  This  will  readily  be  seen  by  a  statement  of  the  method. 

The  values  of  the  planetary  coefficients  A,  S,  etc.,  being  functions  of  the  geocen- 
tric coordinates  of  Venus,  can  be  computed  for  any  assigned  mean  longitude  of  the 
Earth  and  Venus.  They  are  therefore  to  be  computed  for  a  certain  number  of  equi- 
distant values  of  the  mean  longitude  of  each  planet.  For  each  of  these  values 
there  will  be  a  definite  perturbation  of  the  coordinates  of  each  planet,  which  may 
be  computed  and  applied  in  advance.  Thus  the  first  computation  gives  at  once 
numerical  values  of  the  coefficients  in  which  the  effect  of  periodic  perturbation  is 
included.  From  these  are  developed  by  well-known  formulae  the  coefficients  of  the 
sines  and  cosines  of  the  multiples  of  the  mean  longitudes. 

The  perturbations  of  Mars  are  so  small  that  it  was  assumed  that  undisturbed 
values  of  the  coefficients  would  suffice.  But  the  same  method  was  used  owing  to 
its  simplicity  in  theory. 

In  the  case  of  Jupiter  the  analytic  development  would  not  have  involved  the 
difficulty  which  I  have  pointed  out.  But  it  was  so  convenient  to  apply  the  numer- 
ical method  that  it  was  adopted  for  this  planet  also. 

The  action  of  Saturn  is  so  minute  that  a  very  simple  development  suffices.  It 
was  therefore  unnecessary  to  employ  the  numerical  method  in  this  case. 


COEFFICIENTS   FOR  DIRECT  ACTION. 


41 


A.  ACTION  OF  VENUS. 

§  32.  We  shall  now  show  how  the  computations  were  arranged  in  the  case  of 
Venus.  Let  us  first  suppose  that  the  orbits  of  both  planets  are  circular.  Then 
assume  the  Earth  to  be  in  zero  of  longitude.  We  assign  in  succession  60  equidistant 
longitudes  to  Venus,  6°  apart.  For  each  of  these  positions  we  compute  the  values 
of  the  four  principal  coefficients.  Numerical  induction  from  these  special  values 
will  then  give  the  values  of  A,  B,  etc.,  in  a  series  proceeding  according  to  the  cosines 
of  the  multiples  of  the  differences  of  the  mean  longitudes. 

Now  assign  to  the  Earth  a  mean 
longitude  equal  to  any  multiple  of  6°. 
If  we  start  with  Venus  at  inferior  con- 
junction we  shall  have  the  same  series 
of  values  of  the  coefficients  as  before, 
provided  that  we  now  take  the  line 
joining  the  Sun  and  Earth  as  the  axis  of 
X.  Supposing  all  our  coordinates  re- 
ferred to  this  axis  we  should  then  have 
A,  J3,  etc.,  developed  according  to 
cosines  of  multiples  of  the  difference 
of  the  mean  longitudes. 

It  follows  that  in  the  actual  case  of 
the  two  orbits  having  a  small  eccen- 
tricity and  inclination  the  other  terms 
which  we  require  will  be  of  the  order  of 
magnitude  of  these  quantities  and  will  therefore  be  smaller  than  these  principal 
terms.  It  is  therefore  not  necessary  to  divide  the  circle  into  so  many  parts  in  order 
to  obtain  them. 

The  actual  process  was  to  take  the  direction  of  the  solar  perigee  for  1800  as  the 
initial  line,  or  axis  of  X.  The  way  in  which  the  coordinates  were  defined  will  then 
be  seen  by  the  diagram.  Here  on  the  left,  ir'  marks  the  position  of  the  Earth's 
perihelion.  The  positive  direction  of  X  passes  through  the  Sun  and  is  therefore 
directed  toward  the  solar  perigee.  The  Earth  being  in  this  (fixed  position,  the 
coordinates  of  Venus  are  computed  for  60  equidistant  values  of  the  mean  longitude 
of  Venus  differing  by  increments  of  6°.  The  initial  or  zero  value  corresponds  to 
the  mean  inferior  conjunction  of  Venus,  marked  o  in  the  figure,  which  determines 
all  the  other  values;  a  few  of  the  others  are  numbered  in  order. 

For  each  of  these  mean  longitudes,  the  actual  coordinates  of  Venus,  including 
the  effect  of  perturbations  by  the  Earth,  were  computed.  The  position  of  the  Earth 
at  TT',  corresponding  to  the  6°  positions  of  Venus,  was  then  corrected  in  each  of  the 
60  cases  by  the  periodic  perturbations  due  to  each  position  of  Venus.  With 
these  coordinates  60  numerical  values  of  the  ^4 -coefficients  are  computed.  I 


Arrangement  of  Coordinate  Axes,  in  Systems  o,  I,  etc., 
for  Venus. 


42  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

designate  this  system  of  60  values,  corresponding,  perturbations  aside,  to  one 
position  of  the  Earth,  by  the  number  o;  and  I  distinguish  the  values  by  60  indices 
o,  i,  2,  ...  59. 

In  the  next  system,  called  system  i,  the  Earth  has  moved  through  30°  of  mean 
longitude,  or  mean  anomaly,  to  the  position  E±.  The  set  of  60  heliocentric  coordi- 
nates of  Venus  to  be  used  will  be  the  same  as  before,  except  lor  the  perturbations, 
which  will  now  be  those  for  JSlt  or  for  ^-=30°.  But  the  position  corresponding 
to  the  inferior  conjunction  in  this  system  will  be  that  corresponding  to  the  index  5 
in  system  o.  A  new  axis  of  ^  is  now  adopted,  again  passing  through  the  mean 
Sun,  and  therefore  making  an  angle  of  30°  with  the  initial  axis.  The  coordinates 
of  Venus  are  all  transformed  to  this  axis,  and  another  set  of  60  values  of  the 
yl-coefficients  are  computed. 

The  remainder  of  the  process  consists  in  assigning  to  the  mean  longitude  of  the 
Earth  successive  increments  of  30°  until  it  is  brought  around  to  the  position  EIV 
in  mean  anomaly  330°.  In  each  case  the  axis  of  X  is  taken  to  pass  through  the 
mean  Sun. 

From  these  720  special  values  of  the  ^-coefficients  the  general  values  are  sepa- 
rately developed  for  each  of  the  12  systems.  Then  the  general  development  for  any 
system  is  effected  by  a  second  quadrature.  The  final  result  will  be  the  values  of 
Ay  By  etc.,  referred  to  an  axis  always  passing  through  the  mean  Sun. 

Were  we  to  adopt  a  fixed  system  of  coordinate  axes,  it  would  now  be  necessary 
to  transform  these  values  referred  to  the  moving  axis,  to  the  adopted  fixed  system. 
But  the  necessity  of  this  transformation  is  avoided  by  referring  all  the  coordinates, 
those  of  the  Moon  as  well  as  of  the  planet,  to  the  mean  Sun  from  the  beginning. 
This  is  fully  as  simple  as,  perhaps  even  simpler  than,  referring  them  to  a  fixed  axis. 
The  ease  of  doing  it  is  all  the  greater  from  the  fact  that,  in  the  actual  computation  of 
the  lunar  coordinates,  they  are  first  referred  to  the  mean  Moon.  The  transformation 
from  the  mean  Moon  to  the  mean  Sun  is  probably  simpler  than  the  transformation 
to  a  fixed  axis. 

§  33.  Development  of  the  A-coefficients  for  Venus. 

The  computations  relating  to  Venus  are  shown  in  tabular  form  in  Tables  I-  VIII, 
and  will  now  be  explained.  To  obtain  the  12  undisturbed  values  of  the  Sun's 
coordinates,  we  derive  the  equation  of  the  centre  and  the  logarithm  of  the  radius 
vector  from  the  tables  of  the  Sun  found  in  Astronomical  Papers,  Vol.  VI. 

For  the  argument  of  mean  anomaly  of  the  Sun  the  initial  value  is 


corresponding  to  g'  =  o°.     The  increment  for  each  30°  is 

b.M  =30.43830 
resulting  in  the  value  188.0000  for^-'  =  180°. 


COEFFICIENTS  FOR  DIRECT  ACTION.  43 

With  the  12  values  of  M  thus  found  are  taken  the  equation  of  the  centre,  £,  for 
1800,  and  log  r'.  Then 

x'  =  r1  cos  E  y'  =  r'  sin  E 

The  resulting  values  of  x'  and  y'  are  shown  in  Table  I. 
For  Venus,  we  have 

Initial  mean  longitude  =  99°  30'  7" 

this  being  the  longitude  of  the  Earth's  perihelion  for  1800.  .  For  the  same  epoch 

we  have 

Longitude  of  perihelion  of  Venus  =  128°  45'  i7".4 

Initial  mean  anomaly  of  Venus      =  330°  44' 50". 2 

To  find  the  tabular  argument  corresponding  to  this  mean  anomaly  we  proceed  thus: 
Adding  5  increments  of  6°,  we  have 

Mean  anomaly  of  Venus  for  index  5  =  o°  44' 50". 25 
For  this  mean  anomaly  the  precepts  of  Tables  of  Venus,  pp.  278-279,  give 

Tabular  Arg.  K  ;    K^  =  1. 11601 
Increment  of  K  ior  6°  =  3.745014 

We  now  add  one  period  to  A' and  subtract  5  increments 

K&=        1. 11601 

P  =  224.70084 
225.81685 

5  increments  18.72507 
Initial  K,  207.09178 

which  corresponds  to  the  inferior  conjunction  of  Venus  in  system  o. 

The  resulting  values  of  K  are  found  in  Table  II.  With  the  values  of  K  thus 
formed  the  equation  of  the  centre  and  log  r  in  the  elliptic  orbit  of  1800  are  taken 
from  the  tables. 

The  data  lor  the  rectangular  coordinates  are: 

Node  of  Venus,  1800  H=    74°  52'  48". 75 

Perihelion  of  Earth  -IT  =    99   30     7   .6 

Node  referred  to  Perihelion   #=335    22  41   .2 
Inclination  for  1800  7  =      3    23  33  -45 

The  values  of  the  coordinates  x,  y,  and  z  in  the  initial  system  are  now  computed 
by  the  formulae 

u  =  Eq.  Cent,  -f  24°  37'  i8".8o  m  sin  M '=  cos  7  sin  0  m  cos  M=  cos  6 

m'  sin  M'  =  sin  0  m'  cos  M'  =  cos  7  cos  0 
leading  to 

Jl/=24035'i".84  ^'  =  24°39'35".9i 

log  m  =  9.9998680  log  m'  =  9.9993706 

x  =  ntr  cos  (M  +  «)  y  =  m'r  sin  (M1  -fa)  z  =  r  sin  7  sin  u 


44  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

Designating  the  systems  by  suffixes,  and  putting  c  =  cos  30°,  these  coordinates 
were  transformed  to  the  axes  of  the  other  1 1  systems  by  the  formulae 

xi =  cx<>  T  jjXo  y\ =  cy<>    %xv 


and  then,  in  general, 

xn ==    x*-t  y* =    j»-6 

§  34.  Explanation  of  the  tables.  The  periodic  perturbations  of  the  longitudes 
of  the  Earth  and  Venus,  and  of  the  logarithms  of  their  radii  vectores,  omitting 
terms  of  long  period,  are  now  to  be  found. 

TABLE  III:  Mutual  periodic  perturbations. 

For  the  perturbations  of  Venus  by  the  Earth,  Su  and  8/>,  the  arguments  of  the 
double  entry  Tables  VIII  and  XVII  are: 

Hor.  Arg.  g  =  K—  od.65o  =  206^44  +  3.745* 
Vert.  Arg.  II  for  System  o  and  /=  o,  104.35 
Increment  of  II  for  each  system  All  =  20 

"          "  "    "     "      index  A2II  =  — 2.461 

For  the  single  entry  Tables  XI  and  XX  we  have 

Arg.  A  =  i.62203(£--£-') 

For  the  index  i        g—  330°. 75  +  6°?' 

For  the/th  system  g'  =  30 °j 
Hence,  for  /  =  o,  j  =  o,   Arg.  A  =  536.49 
Increment  for  each  unit  of  t,    &A  =  -f  9.732 
"          "      "       "     "  j,  AM  =  —  48.661 

With  the  values  of  the  arguments  thus  formed  the  periodic  perturbations  of  Venus 
by  the  Earth  are  taken  from  the  Tables  VIII,  XVII,  XI,  and  XX. 

For  the  corresponding  perturbations  of  the  Earth  by  Venus,  we  have 

Hor.  Arg.  £-=30.43837 
Vert.  Arg.  II  for/=  o;  t  =  o  •  •  •  165.375 
Increment  for  each  unit  of  /;  JA_§-=  3 
"      "       "     «/;   -24.383 

Argument  A  has  the  same  value  as  in  the  Tables  of  Venus. 


COEFFICIENTS  FOR  DIRECT  ACTION.  45 

The  perturbations  of  the  longitude  and  log.  radius  vector  of  the  Earth  found  with 
these  values  of  the  arguments  are  given  in  the  columns  $v'  and  8p. 

TABLE  IVa  AND  IV£.  The  perturbations  in  Table  III  are  transformed  into 
increments  of  the  rectangular  coordinates  of  Venus  and  the  Earth. 

Neglecting  the  cosine  of  the  inclination  we  have  for  Venus  when  referred  to  the 
initial  system  of  axes 

AA;O  =  —  y  sin  i"Su  -f  x&p  Ay0  =  x  sin  i"Bu  +  ySp 

the  tabular  8p  being  multiplied  by  the  modulus  of  logarithms.  For  the  other 
systems  the  transformation  is  made  by  the  formulas  for  the  transformation  of  the 
coordinates  themselves.  The  results  are  given  in  full,  in  units  of  the  8th  place  of 
decimals,  in  Table  IV.  Applying  them  to  the  undisturbed  coordinates,  we  have 
the  coordinates  of  Venus  for  each  position  of  the  two  bodies. 

TABLE  V.  The  values  of  the  solar  coordinates  in  Table  I,  of  the  Venus  coor- 
dinates in  Table  II,  after  being  transformed  to  the  axis  of  the  system,  and  of  the 
increments  in  Table  IV,  are  added  so  as  to  form  the  disturbed  geocentric  coordi- 
nates of  Venus  in  each  system  for  each  position  of  Venus. 

TABLE  VI.  With  the  perturbations  of  latitude  in  the  different  systems  the  dis- 
turbed geocentric  coordinate  Z  was  computed  and  tabulated. 

With  these  geocentric  coordinates  are  computed  the  720  values  of  the  four 
coefficients  A,  Z?,  C,  and  D  defined  in  §  7.  Since 

A  +  £  + C=o 

the  computation  of  C  might  have  been  dispensed  with.  It  was,  however,  carried 
through  as  an  additional  check  on  the  accuracy  of  the  work.  The  latter  was,  how- 
ever, done  in  duplicate,  the  check  being  incomplete. 

TABLE  VII  gives  the  values  of  the  coefficients  thus  computed. 

The  coefficients  E  and  Flead  to  appreciable  inequalities  only  in  the  case  of  the 
argument  0,  and  have  been  treated  separately.  Their  special  values  were  computed 
for  six  systems  and  thirty  indices  only,  and  are  found  in  Table  VIII. 

§  35.  The  process  of  developing  the  general  value  of  each  coefficient  in  a  periodic 
series  is  given  by  Briinnow  in  his  S-ph'drischen  Astronomic,  Taking  A  as  an 
example  we  first  develop  the  value  for  each  system  in  the  form 

Ak  =  '£  (ak  cos  iL  +  bk  sin  t'L) 

where  k  is  the  number  of  the  system  and  L  the  difference  of  the  mean  longitudes 
of  Venus  and  the  Earth, 

L-v-f 

We  thus  have  12  values  of  each  of  the  coefficients  ak  and  bk,  one  corresponding 
to  each  value  of  g' .  These  values  are  then  again  developed  in  the  form 

'  +  bk<j  s'mjg-')  bh  =  2  (a*,/  cos  ig1  +  bki 


44  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

Designating  the  systems  by  suffixes,  and  putting  c  =  cos  30°,  these  coordinates 
were  transformed  to  the  axes  of  the  other  1 1  systems  by  the  formulae 


and  then,  in  general, 

§  34.  Explanation  of  the  tables.  The  periodic  perturbations  of  the  longitudes 
of  the  Earth  and  Venus,  and  of  the  logarithms  of  their  radii  vectores,  omitting 
terms  of  long  period,  are  now  to  be  found. 

TABLE  III:  Mutual  periodic  perturbations. 

For  the  perturbations  of  Venus  by  the  Earth,  8u  and  8p,  the  arguments  of  the 
double  entry  Tables  VIII  and  XVII  are: 

Hor.  Arg.  g  =  K  —  od.65o  =  2o6rf.44  +  3-745* 
Vert.  Arg.  II  for  System  o  and  i=  o,  104.35 
Increment  of  II  for  each  system  All  =  20 

"          "  "    "     "      index  AJI  =  —  2.461 

For  the  single  entry  Tables  XI  and  XX  we  have 

For  the  index  i        g=  330°. 75  +  6°/' 

For  the/th  system  g'  =  30  °/ 
Hence,  for  /  =  o,  j  =  o,  Arg.  A  =  536.49 
Increment  for  each  unit  of  i,    &.A  =  +  9.732 
"          "       "       "     "  _/,  AM  =  —  48.661 

With  the  values  of  the  arguments  thus  formed  the  periodic  perturbations  of  Venus 
by  the  Earth  are  taken  from  the  Tables  VIII,  XVII,  XI,  and  XX. 

For  the  corresponding  perturbations  of  the  Earth  by  Venus,  we  have 

Hor.  Arg.  £•=  30.43837 
Vert.  Arg.  II  for/  =  o  ;  *  =  o  •  •  •  165.375 
Increment  for  each  unit  of  / ;  JA<g'=  3 
"  "      "       "     "  /;   —  24.383 

Argument  A  has  the  same  value  as  in  the  Tables  of  Venus. 


COEFFICIENTS  FOR  DIRECT  ACTION.  45 

The  perturbations  of  the  longitude  and  log.  radius  vector  of  the  Earth  found  with 
these  values  of  the  arguments  are  given  in  the  columns  §v'  and  Sp'. 

TABLE  IVa  AND  IV£.  The  perturbations  in  Table  III  are  transformed  into 
increments  of  the  rectangular  coordinates  of  Venus  and  the  Earth. 

Neglecting  the  cosine  of  the  inclination  we  have  for  Venus  when  referred  to  the 
initial  system  of  axes 

A#0  =  —  y  sin  i"Su  +  xSp  Ay0  =  x  sin  i"Su  -f-  ySp 

the  tabular  8p  being  multiplied  by  the  modulus  of  logarithms.  For  the  other 
systems  the  transformation  is  made  by  the  formulae  for  the  transformation  of  the 
coordinates  themselves.  The  results  are  given  in  full,  in  units  of  the  8th  place  of 
decimals,  in  Table  IV.  Applying  them  to  the  undisturbed  coordinates,  we  have 
the  coordinates  of  Venus  for  each  position  of  the  two  bodies. 

TABLE  V.  The  values  of  the  solar  coordinates  in  Table  I,  of  the  Venus  coor- 
dinates in  Table  II,  after  being  transformed  to  the  axis  of  the  system,  and  of  the 
increments  in  Table  IV,  are  added  so  as  to  form  the  disturbed  geocentric  coordi- 
nates of  Venus  in  each  system  for  each  position  of  Venus. 

TABLE  VI.  With  the  perturbations  of  latitude  in  the  different  systems  the  dis- 
turbed geocentric  coordinate  Z  was  computed  and  tabulated. 

With  these  geocentric  coordinates  are  computed  the  720  values  of  the  four 
coefficients  A,  B,  C,  and  D  defined  in  §  7.  Since 

A  +  B  + C=o 

the  computation  of  C  might  have  been  dispensed  with.  It  was,  however,  carried 
through  as  an  additional  check  on  the  accuracy  of  the  work.  The  latter  was,  how- 
ever, done  in  duplicate,  the  check  being  incomplete. 

TABLE  VII  gives  the  values  of  the  coefficients  thus  computed. 

The  coefficients  E  and  F  lead  to  appreciable  inequalities  only  in  the  case  of  the 
argument  6,  and  have  been  treated  separately.  Their  special  values  were  computed 
for  six  systems  and  thirty  indices  only,  and  are  found  in  Table  VIII. 

§  35.  The  process  of  developing  the  general  value  of  each  coefficient  in  a  periodic 
series  is  given  by  Briinnow  in  his  Spharischen  Astronomic.  Taking  A  as  an 
example  we  first  develop  the  value  for  each  system  in  the  form 

Ali  =  'Z  (ak  cos  iL  +  bh  sin  iL) 

where  k  is  the  number  of  the  system  and  L  the  difference  of  the  mean  longitudes 

of  Venus  and  the  Earth, 

Z  =  v-£-' 

We  thus  have  12  values  of  each  of  the  coefficients  ak  and  bk,  one  corresponding 
to  each  value  of  g'.  These  values  are  then  again  developed  in  the  form 

'  bu  =  2  (<**,/  cos  ig'  +  bk  ' 


46  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

These  being  substituted  in  the  general  expression  given  above  for  Ak  gives  the 
value  of  A  itself  in  the  form 

A  =  22  [>  cos  (iL  +jg')  +  b  sin  (iL  +jg')] 

The  development  was  effected  in  this  way  up  to  i  =  8  only,  this  being  the  limit 
for  possible  sensible  terms  other  than  the  Hansenian  term  of  long  period  depending 

on  the  argument 

2g'  -g 


§  36.  The  Hansenian  Venus-term  of  long  period.  The  computation  of  this 
inequality  requires  the  determination  of  the  coefficients  for  i  =  18,  which  we  obtain 
trom  the  general  formulae  thus.  Putting,  in  any  one  system, 

AH  Alt  Av  •  •  •  AM 
for  the  60  values  of  A,  and 

A  cos  i8Z  +  A  sin  i8Z 


for  the  pair  of  terms  depending  on  the  argument  i8Z,  the  general  formulae  give 


A0  +  Al  cos  108°  +  A2  cos  216°  +  .  •  • 
T>oAi  =  Al  sin  108°  +  At  sin  216°  +  •  •  • 

the  angles  increasing  by  108°  in  each  term.  The  fifth  angle  will  be  180°  -j-  2ir, 
so  that  the  only  numerically  different  values  of  the  coefficients  which  enter  into 
the  series  besides  i  and  o  are 

sin  18°,  cos  18°,  sin  36°,  and  cos  36° 
For  example,  we  have 


Atl  —  Al  sin  18°  —  At  cos  36°  +  A3  cos  36°  +  A<  sin  18°  +  •  •  • 
30^4,  =  Al  cos  18°  —  At  sin  36°  —  A3  sin  36°  +  AI  cos  18°  +  •  •  • 

From  the  cyclic  order  of  the  coefficients  the  method  of  computing  Ac  and  A, 
is  as  follows: 

With  the  60  values  of  any  one  coefficient,  say  A,  in  any  one  system, 

A0,  Alt  Av  •  •  •,  AM 
compute 

'  =  A 


0 
= 


Alt  +  Ait 


Next: 


Next: 


COEFFICIENTS   FOR  DIRECT  ACTION. 


J'-Aj-AJ        A>'=A,'-AS> 


47 


We  then  have,  in  each  system 
2oAc=A0"  +  ACils\ni8°  +  A 


_  .  .  OQ  A  *  £.Q 

with  similar  values  for  B,  C,  and  D. 

The  numerical  results  of  these  processes  for  each  system  are  shown  in  Table  IX. 

The  next  step  is  to  develop  each  set  of  numerical  values  of  any  one  pair  of  coeffi- 
cients, say  Ac,  and  A,  in  the  form 


30^4c  =  «,,  +  «,  cos  g'  +  a2  cos  2g'  +  ftl  sin  g'  +  ftt  sin  2g' 
3oAt  =  ag'  +  a/  cos  g'  +  atf  cos  2g'  +  /3/  sin  g'  +  ft2'  sin  2g' 

These  are  to  be  substituted  in  the  general  form 

A  —  Ae  cos  i8L  +  At  sin  i8L 
Retaining  only  terms  which  may  be  wanted  for  our  purpose,  we  shall  have 

30^4  =  a0  cos  i8L  +  <*„'  sin  i8L 

+  1  (a,  -  £/)  cos  (iSL  +  g')  +  \  (a/  +  /3J  sin  (i8L  +  g') 
+  i(a,  -  /8,')  cos  (i8Z  +  2P-')  +  i(a,'  +  ft.)  sin  (i8Z,  +  2^') 


(«) 


+  K«3  -  ft')  cos  (i8Z  +  3£-')  +  HO,'  +  ft)  sin  (iSL  +  Zg') 

TABLE  I. 

SUN'S  GEOCENTRIC  COORDINATES  IN  THE  MEAN  ORBIT  OF  1800, 
REFERRED  TO  MEAN  SUN  AS  DIRECTION  OF  Axis  OF  X. 


System. 

g' 

x' 

/ 

o 
i 

o 

o 
30 

+0.983  2075 
+0.985  3853 

O.OOOOOOO 

+0.016  8542 

2 

60 

+0.991  3897 

+0.029  1452 

3 

90 

+0.0997183 

+0.033  5823 

4 

1  20 

+1.008  1877 

+0.029  0233 

5 

ISO 

+1.0144741 

+0.016  7321 

6 

180 

+1.0167929 

O.OOOOOOO 

7 

2IO 

+1.0144741 

—  0.016  7321 

8 

24O 

+1.0081877 

—0.029  0233 

9 

27O 

+0.999  7183 

—0.033  5823 

10 

300 

+0.991  3897 

—0.029  1452 

ii 

330 

+0.985  3853 

—  0.016  8542 

ACTION   OF  THE   PLANETS  ON   THE   MOON. 


TABLE   II. 

COMPUTATION  OF   RECTANGULAR  COORDINATES  OF  VENUS    IN   THE   ELLIPTIC   ORBIT  OF 
l8OO,  REFERRED  TO  SOLAR  PERIGEE  AS  AXIS  OF  X. 


i 

Arg.  K. 

Eq.  Cent. 

log.  r 

log.  X 

\og.y 

log.  Z 

o 

207.0918 

/   n 
—23  15.02 

9.856  7321 

—0.856  5920 

+7.7269130 

+8.242  1742 

i 

210.8368 

—  18  47.49 

9.856  5914 

—9.8542879     —8.8490901 

+8-33I  7492 

2 

214-5818 

—  14   7-30 

9.8564812 

—9.8470688     —9.1638652 

+8401  8640 

3 

218.3268 

—  9  17.60 

9.856  4028 

—9.834  7638 

—9.341  2292 

+84579729 

4 

222.0718 

—  4  21.63 

9-8563570 

—9.8170717 

-9.463  1513 

+8.503  3027 

5 

1.1160 

+  o  37.28 

9.8563445 

—9-793  5309     —9-554  3215 

+8.539  9034 

6 

4.8610 

+  5  35.76 

9-856  3653 

—9-763  4657     —9-625  5304 

+8.569  1404 

7 

8.6060 

+  10  30.48 

9.8564193 

—  9.725  9000     —  9.682  4522 

+8.591  9486 

8 

12.3510 

+  15  18.10 

9-856  5057 

—9.679  4042     —9-728  4252 

+8.608  9752 

9 

16x1961 

+  19  55-43 

9.856  6236 

—9.621  8221     —9.765  5624 

+8.6206617 

10 

19.841  1 

+24  19.34 

9-8567717 

—9-549  7350     —9-795  2686 

+8.627  2934 

ii 

23.5861 

+28  26.93 

9.8569481 

—9457  3022     —9.818  5057 

+8.629  0280 

12 

27.3311 

+32   15-39 

9-857  1509 

—9-333  3365     —9.835  9401 

+8.625  9108 

13 

31-0761 

+35  42.29 

9-857  3778 

—9.  1  5  1  9096     —9.848  0289 

+8.6178781 

14 

34-8211 

+38  45-29 

9.857  6261 

—8.824  3320     —9-855  0707 

+8.6047511 

IS 

38.5661 

+41  2244 

9-8578931 

+7-961  5592     -9-857  2356 

+8.5862194 

16 

42.3112 

+43  32.07 

9-858  1756 

+8.929  0946     —9-854  5813 

+8.561  8070 

17 

46.0562 

+45  12.78 

9.858  4706 

+9.203  5288     —9-847  0574 

+8.530  8192 

18 

49.8012 

+46  23.54 

9-858  7747 

+9.367  1085     —9-834  5015 

+8492  2492 

19 

53.5462 

+47   3-66 

9.859  0846 

+9.482  0030     —  9.816  6224 

+8.444  6187 

20 

57.2912 

+47  12.78 

9.859  3968 

+9.5688886     —9.7929712 

+8.385  6881 

21 

61.0362 

+46  50.89 

9.859  7079 

+9.637  21  13       —9.762  8896 

+8.3118851 

22 

64.7812 

+45  58.29 

9.860  0146 

+9.692  0696 

—9-725  4250 

+8.2170551 

23 

68.5263 

+44  35-65 

9.8603134 

+9.7365149 

—9.679  1841 

+8.0892497 

24 

72.2713 

+42  43.92 

9.8606011 

+9.772  5022 

—9.6220691 

+7.900  1224 

25 

76.0163 

+40  24.44 

9.8608748 

+9.801  3428 

-9-550  7649 

+7.5469461 

26 

79.7613 

+37  38.73 

9.861  1314 

+9.823  9406 

—9.459  6400 

—6.971  8246 

27 

83.5063 

+34  28.65 

9.861  3684 

+9.840  9243 

—9.338  oooi 

—7-731  3881 

28 

87-2513 

+30  56.31 

9.861  5830 

+9.852  7264 

—9.161  4423 

—7.9903231 

29 

90.9963 

+27   4.01 

9.861  7733 

+9.859  6294 

-8.8495051 

—8.148  1795 

30 

94.7414 

+22  54.28 

9.861  9370 

+9.861  7933 

+7-6393795 

—8.260  1046 

31 

98.4864 

+  18  29.84 

9.862  0726 

+9.859  2710 

+8.8997388 

—8.345  1900 

32 

102.2314 

+  13  53-52 

9.862  1786 

+9.8520116 

+9.1862719 

—8.412  3233 

33 

105.9764 

+  9   8.29 

9.862  2540 

+9.8398567 

+9.354  2638 

—8.466  3544 

34 

109-7214 

+  4  17.19 

9.862  2779 

+9-822  5254 

+9-471  5502 

—8.5102060 

35 

113.4664 

—  o  36.64 

9.862  3009 

+9-799  5863 

+9.5600170 

—8.545  7583 

36 

117.2114 

—  5  30.07 

9.862  2899 

+9.770  4104 

+9.629  5228 

—8.574  2757 

37 

120.9564 

—  10  20.01 

9.862  2381 

+9.734  0928 

+9.6853431 

—8.596  6294 

38 

124.7015 

-15   3-29 

9.862  1550 

+9.6893188 

+9-7306171 

—8.613  4244 

39 

128.4465 

—19  36.93 

9.862  0415 

+9.634  1256 

+9-7673464 

—8.625  0747 

40 

132.1915 

—23  57-o8 

9.8618988 

+9-565  4464 

+9-7968687 

-8.631  8465 

41 

135.9365 

-28   3-60 

9.861  7283 

+0.478  1499 

+9.82O  1022 

-8.6338851 

42 

139.6815 

—31  5i-i6 

9-861  5318 

+9.362  7126 

+9.8376839 

—8.631  2325 

43 

143.4265 

-35  18.17 

9.8613113 

+9.1982317 

+9.85O  O5OO 

—8.623  8210 

44 

I47.I7I5 

-38  22.35 

9.861  0693 

+8.921  4921 

+9.8574837 

-8.6114784 

45 

150.9166 

—41   1.70 

9.8608081 

+7.912  5788 

+9.860  1443 

—8.593  9051 

46 

154.6616 

—43  14.42 

9.860  5307 

—8.8273540 

+9-858o8l7 

—8.570  6520 

47 

158.4066 

—44  58.99 

9-860  2398 

—9.1518035 

+9.851  2413 

—8.541  0589 

48 

162.1516 

—46  14.23 

9.8599388 

-0-3323188 

+9.8394597 

—8.504  1837 

49 

165.8966 

—46  59.22 

9-859  6307 

—9-455  9020 

+9.822  4495 

—8.458  6539 

50 

169.6416 

—47  13.42 

9.8593190 

—9.548  1734 

+9-7997714 

—8.402  4145 

51 

173-3866 

—46  56.56 

9.8590071 

—9.620  2216 

+9.770  7859 

—8.332  2360 

52 

I77.I3I7 

—46   8.76 

9.8586983 

—9.6778438 

+9-734  5737 

—8.242  6677 

53 

180.8767 

—44  50.45 

9.8583962 

—9.724  4363 

+9-689  7900 

—8.1234525 

54 

184.6217 

-43   2.43 

9.858  1040 

—9.762  1421 

+9.634  4660 

—7-951  5304 

55 

188.3667 

—40  45.82 

9.857  8250 

-9-792  3808 

+9.565  4503 

—7-653  3423 

56 

192.1117 

-38   2.06 

9.857  5624 

—9.816  1217 

+9.477  5085 

—4.049  3707 

57 

195-8567 

—34  52.90 

9.8573192 

—9.834  0339 

+9.360  8616 

+7.6516300 

58 

109.6017 

—31  20.43 

9.8570981 

—9.846  5740 

+9-I93  8645 

+7-950  6732 

59 

203.3468 

—27  26.94 

9.8569016 

—9.854  0381 

+8.9098324 

+8.1228613 

COEFFICIENTS  FOR  DIRECT  ACTION. 


49 


TABLE   III. 
MUTUAL  PERIODIC  PERTURBATIONS  OF  VENUS  AND  THE  EARTH. 

The  term  of  long  period  is  omitted.     The  tabular  unit  is  cf'.oi  in  <!«  and  dvf,  and  10— 8  in  Sp  and 


i 

System  o. 

System  i. 

System  2. 

8u 

8v'    dp 

dp' 

Su 

dv' 

dp 

dp' 

du 

dv' 

dp 

8P' 

o 

+  239 

—169   +  347 

+464 

+  534 

-390 

+  32i 

+472 

+  543 

—402 

+  229 

+537 

I 

+  65 

—151   +  394 

+440 

+  432 

—423 

+  398 

+424 

+  545 

—507 

+  331 

+467 

2 

—  146 

—  116   +  515 

+365 

+  267 

—423 

+  540 

+334 

+  479 

—573 

+  5H 

+349 

3 

—  406 

-  65   +  675 

+254 

+  23 

-379 

+  7i6 

+214 

+  315 

—590 

+  724 

+203 

4 

-  705 

+  4   +814 

+  126 

-  295 

—303 

+  880 

+  78 

+  58 

-558 

+  910 

+  42 

5 

—  1018 

+  82   +897 

+  o 

-  662 

—  194 

+  988 

-  62 

-276 

-477 

+  1045 

-"7 

6 

—1304 

+  158   +  898 

—  Ill 

—1052 

—  62 

+1023 

—  194 

—  659 

—357 

+  1093 

—261 

7 

—1541 

+219   +  823 

—  206 

—  1420 

+  80 

+  941 

-308 

-1058 

—204 

+  1044 

-388 

8 

—1702 

+264  ,  +  687 

—284  !  —1728 

+221 

+  779 

-397 

—  1435 

—  27 

+  895   -4»9 

9 

-1796 

+289   +  SOI 

—352 

-1941 

+348 

+  551 

—459 

—  1760 

+  161 

+  658 

-562 

10 

—1820 

+301  ;  +  281 

—414 

—2049 

+449 

+  273 

—495 

-1991 

+348 

+  349 

—602 

n 

—1777 

-j-307   +  26 

-468 

—2047 

+522 

—   21 

-5H 

—2103 

+517 

+   5 

—  612 

12 

—1674 

+313   —  234 

—510 

—  1950 

+566 

-316 

-518 

—2089 

+657 

-  343 

—594 

13 

-1508 

+323   —  5o6 

-538 

-1767 

+589 

-  596 

-515 

-1956 

+757 

—  661 

—557 

14 

-!286 

+343   —  767 

—549 

-1517 

+599 

-851 

—504 

—  1721 

+818 

-  935 

—506 

15 

—1013 

+370   —1002 

—544 

—  i2ii 

+605 

-1066 

-485 

—1417 

+847 

—  1146 

—447 

16 

-  695 

+403    —1205 

—524 

-  857 

+607 

—  1236 

—454 

—  1057 

+850 

—  1298 

-387 

17 

—  350 

+446 

—  I36l 

-486 

-  475 

+610 

-1348 

-406 

-  666 

+836 

-1386 

—322 

18 

+  19 

+498 

—  1462 

—433 

—  77 

+611 

—  1399 

—346 

-  260 

+812 

—  1407 

—255 

19 

+  407 

+559 

—  I486 

—353 

+  320 

+613 

-1382 

-269 

+  H9 

+78i 

—  1357 

—  181 

20 

+  787 

+622 

—  1417 

-247 

+  70i 

+614 

—  1302 

-178 

+  535 

+741 

—  1236 

—  99 

21 

+1139 

+681   —1244 

-113 

+1047 

+616 

—  1148 

—  72 

+  883 

+695 

—1053 

—  6 

22 

+1431 

+723   —  966 

+  40 

+I35I 

+617 

—  921 

+  So 

+  1182 

+643 

—  825 

+  96 

23 

+1629 

+740   —  597 

+205 

+1586 

+611 

—  6l9 

+  189 

+  1417 

+586 

—  541 

+209 

24 

+1709 

+726   —  180 

+367 

+1725 

+590 

—  248 

+338 

+  1578 

+524 

—  226 

+328 

25 

+1660 

+682   +  263 

+5i6 

+1748 

+550 

+  164 

+489 

+  1652 

+458 

+  121 

+453 

26 

+1489 

+606   +  694 

+646 

+1636 

+484 

+  586 

+630 

+  1623 

+3& 

+  488  ;  +578 

27 

+  1215 

+504   +1074 

+752 

+  1394 

+395 

+  973 

+750 

+  1478 

+288 

+  846 

+695 

28 

+  854 

+37<5   +1381 

+836 

+  1045 

+285 

+  1287 

+844 

+  1212 

+179 

+  1162 

+799 

29 

+  438 

+226   +1583 

+890 

+  616 

+160 

+  1503 

+GOO 

+  842 

+  56 

+  1399 

+875 

30 

—   9 

+  58   +1667 

+912 

+  148 

+  24 

+  1608 

+922 

+  397 

-  77 

+  1532 

+918 

3i 

—  451 

—122   +  1616 

+898 

—  321 

—  122 

+  1596   +909 

—  79 

—214 

+  1544 

+923 

32 

-  859 

—301  +1439 

+843   -  764 

—  272 

+  1463 

+863 

—  549 

-342 

+  1437 

+887 

33 

—  1197 

—468   +1142 

+755   —i  145 

—  426 

+  1210 

+784 

-  971 

—467 

+  1213 

+815 

34 

—  1440 

—  610   +  765 

+637 

—1429 

-571 

+  860 

+673 

-1309 

-581 

+  896 

+709 

35 

—1577 

-7i8   +  336 

+502 

-1598 

—7OO 

+  436 

+536 

—  1540 

-684 

+  Soi 

+576 

36 

—  1605 

—793   —  103 

+355 

-1636 

-803 

—  14 

+377 

—  1643 

-777 

+  66   +420 

37 

—  1529 

—833   —  524 

+204 

—  1557 

—870 

-  453 

+206 

—  1610 

-849 

-  378   +247 

38 

—1357 

-837 

—  911 

+  51 

—  1360 

-898 

-  856 

+  33 

-1443 

-893 

—  797 

+  07 

39 

—  lotjS 

-811 

—  1221 

—  102 

-1085 

-889 

-1183 

—132 

—  1163 

-907 

—1148 

—"5 

40 

-  770 

-755 

—  1440 

—248 

—  747 

-844 

—1422 

—284 

-  800 

-883 

—1404 

-288 

41 

-  393 

-675 

-1548 

-380 

—  370 

-770 

-1557 

—418 

-  389 

—824 

-I55i 

—443 

42 

-   8 

-575 

—  1540 

-485 

+  19 

-669 

—  1579 

—532 

+  31 

-733 

-1578 

—573 

43 

+  360 

—467 

—  1430 

—559 

+  395 

—547 

—  1491   —620 

+  426 

—620 

—1503 

-673 

44 

+  681 

—359 

—  1243 

—599 

+  730 

—410 

—  1303   —678 

+  780 

-487 

-1328 

—741 

45 

+  950 

—  264 

—1003 

-608 

+  IOO2 

-269 

—  1035  !  —702 

+  1068 

—340 

—1067 

-775 

46 

+1160 

-187 

—  739 

—593 

+  1192 

—  135 

-  731   -688 

+  1273 

-188 

-  750 

—772 

47 

+  1318 

—  130 

—  463 

-561 

+  1302 

—  20 

—  415   —641 

+  1383 

-  38 

—  403 

—732 

48 

+1424 

—  92 

—  186 

-Si8 

+  1347 

+  68 

-  106   -566 

+  1399 

+  99 

—  59 

—656 

-19 

+1485 

-69 

+   72 

-465 

+  1334 

+  127 

+  M8 

-476 

+  1335 

+212 

+  240 

—552 

50 

+J497 

—  59 

+  304 

-398 

+  1284 

+  155 

+  367 

—374 

+  1212 

+291 

+  472 

—422 

Si 

+1467 

—  61 

+  450 

—319 

+  1206 

+  160 

+  537 

-268 

+  I066 

+331 

+  627 

—277 

52 

+  1401 

-  74 

+  641 

—232 

-j-IIII 

+  148 

+  655 

-163 

+  904 

+336 

+  709 

—129 

53 

+1306 

—  95 

+  739 

-136 

+  1007 

-(-116 

+  719 

-  55 

+  757 

+308 

+  739 

+  14 

54 

+1191 

—  122 

+  783 

-38 

+  903 

+  65 

+  731 

+  52 

+  634 

+254 

+  701 

+  144 

55 

+1050 

—  147 

+  777 

+  63 

+  799 

+  i 

+  688 

+161 

+  535 

+  182 

+  627 

+264 

56 

+  889 

-I65 

+  717 

+  168 

+  728 

-  76 

+  603 

+267 

+  467 

+  92 

+  504 

+371 

57 

+  723 

-175 

+  6n 

+272 

+  670 

-162 

+  494 

+362 

+  443 

-  17 

+  390 

+46l 

58 

+  556 

-179 

+  488 

+367 

+  630 

—251 

+  388 

+435 

+  462 

-142 

+  276 

+527 

59 

+  398 

-177 

+  387   +436 

+  591 

-329 

+  320 

+474 

+  SGI 

-274 

+  212 

+556 

5° 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE  III.— Continued. 
MUTUAL  PERIODIC  PERTURBATIONS  OF  VENUS  AND  THE  EARTH. 

The  term  of  long  period  is  omitted.     The  tabular  unit  is  o".oi  in  6u  and  6v',  and  10— 8  in  6p  and 


i 

System  3. 

System  4. 

System  5. 

3u 

8v' 

ip 

iff 

du 

dv' 

9p 

V 

du 

dv' 

dp 

V 

o 

+  391 

—  295 

+  106 

+639 

+  249 

—  194 

—  20 

+749 

+  110 

-  87 

—  121 

+839 

i 

+  452 

—  431 

+  220 

+555 

+  326 

—  337 

+  89 

+669 

+  223 

-  253 

-   31 

+777 

2 

+  456 

-  543 

+  424 

+419 

+  349 

—  457 

+  293 

+533 

+  273 

—  391 

+  170 

+649 

3 

+  375 

—  619 

+  667 

+251 

+  299 

-  548 

+  547 

+357 

+  245 

—  493 

+  424 

+477 

4 

+  197 

—  647 

+  893 

+  66 

+  170 

—  604 

+  802 

+  156 

+  128 

-  556 

+  665 

-1-277 

5 

-  68 

—  623 

+  1059 

-118 

-  38 

-  621 

+  1008 

-  53 

-  65 

—  579 

+  903 

+  & 

6 

—  393 

—  551 

+  "32 

—287 

—  310 

-  596 

+  1132 

—251 

—  3i4 

-  568 

+  1053 

—  144 

7 

—  744 

—  435 

+  1106 

—436 

—  614 

—  529 

+  H45 

—430 

-  595 

-  524 

+  iiii 

—344 

8 

-1097 

-  283 

+  977 

—557 

—  924 

—  421 

+  1042 

-576 

-  882 

-  447 

+  1062 

—523 

9 

—1424 

—  103 

+  700 

-646 

—  1208 

—  278 

+  838 

-688 

—  1148 

—  339 

+  902 

-672 

10 

—1690 

+  97 

+  463 

-701 

—  1441 

-  108 

+  547 

-761 

—  1362 

—  202 

+  637 

-781 

it 

-1876 

+  305 

+  109 

—720 

—  1604 

+  79 

+  200 

—799 

-1499 

—  44 

+  293 

-845 

12 

-1958 

+  507 

-  271 

-702 

-1688 

+  276 

-  180 

—799 

-1545 

+  126 

—  95 

-863 

13 

—  1919 

+  692 

—  647 

-654 

-1681 

+  476 

-  563 

-765 

—1501 

+  303 

-  489 

-844 

14 

—  1761 

+  843 

—  977 

-578 

-1582 

+  667 

—  921 

-695 

-1372 

+  478 

-  859 

-788 

IS 

-1505 

+  952 

—  1226 

-487 

-1393 

+  839 

—1219 

—594 

-1174 

+  648 

—1171 

-701 

16 

—  i  173 

+  1014 

-1378 

-388 

—  II2O 

+  979 

-1433 

-471 

-  914 

+  805 

—1412 

-588 

17 

-  800 

+  1035 

-1458 

-287 

-  787 

+  1078 

—1541 

-332 

—  610 

+  944 

—  1554 

—449 

18 

-  412 

+  1019 

-1448 

-188 

—  429 

+  1127 

—  1539 

—  194 

—  284 

+  1051 

-1588 

—291 

19 

—  23 

+  080 

-1366 

-  95 

-   6p 

+  1129 

-1431 

-  62 

+  45 

+  II2I 

—  1503 

-125 

20 

+  344 

+  922 

—1219 

—  5 

+  260 

+  1089 

—1245 

+  56 

+  347 

+  "43 

—  1313 

+  37 

21 

+  679 

+  853 

—  1014 

+  84 

+  547 

+  1021 

—  999 

+  162 

+  595 

+  II2O 

—  1036 

+  188 

22 

+  967 

+  774 

-  758 

+  172 

+  788 

+  931 

—  713 

+254 

+  778 

+  1056 

—  706 

+313 

23 

+  "95 

+  682 

-  466 

+262 

+  975 

+  827 

—  403 

+340 

+  893 

+  959 

—  356 

+416 

24 

+  1349 

+  582 

—  I5i 

+353 

+  IIO2 

+  712 

-  81 

+4'9 

+  942 

+  841 

-   7 

+495 

25 

+  1428 

4  477 

+  171 

+446 

+  1161 

+  587 

+  241 

+492 

+  936 

+  710 

+  322 

+557 

26 

+  1427 

+  364 

+  489 

+542 

+1148 

+  449 

+  546 

+558 

+  875 

+  569 

+  623 

+609 

27 

+  1339 

+  248 

+  786 

+635 

+1067 

+  307 

+  817 

+617 

+  765 

+  422 

+  879 

+650 

28 

+  1164 

+  126 

+  1059 

+726 

+  920 

+  163 

+  1043 

+674 

+  608 

+  258 

+  1086 

+684 

29 

+  003 

—   4 

+  1273 

+804 

+  719 

+  13 

+  1208 

+726 

+  412 

+  107 

+  1216 

+707 

30 

+  562 

—  143 

+  1416 

+860 

+  473 

-  138 

+  1313 

+770 

+  194 

-  56 

+1278 

+720 

31 

+  160 

-  288 

+  1454 

+886 

+  186 

—  290 

+  1341 

+803 

-  36 

—  219 

+1263 

+726 

32 

-  273 

—  429 

+  1379 

+876 

-  137 

—  442 

+  1282 

+814 

-  269 

—  379 

+  "77 

+720 

33 

-692 

-  562 

+  "83 

+829 

-  472 

—  591 

+  1117 

+796 

—  491 

-  536 

+  1003 

+707 

34 

—1052 

-675 

+  887 

+743 

-  781 

—  730 

+  854 

+742 

-  692 

-685 

+  773 

+676 

35 

-1325 

-  764 

+  514 

+625 

-1038 

-  848 

+  500 

+652   -  857 

—  824 

+  461 

+624 

36 

-1484 

-832 

+  96 

+479 

—  1206 

-  937 

+  94 

+526   -  966 

-  948 

+  93 

+540 

37 

-1523 

-  878 

—  334 

+3" 

—  1269 

—  991 

-  329 

+376   —1002 

-1047 

-  309 

+425 

38 

-1427 

—  902 

—  751 

+  129 

—  1221 

—  1007 

-  740 

+207   -  945 

—  1106 

-  718 

+281 

39 

—  1212 

—  906 

—  III2 

—  59 

—  I06S 

—  991 

-1095 

+  26   -  797 

—1124 

—  1082 

+  "4 

40 

-  888 

-884 

—  1392 

—244 

—  809 

—  946 

—  1376 

—  158   —  564 

—1095 

—  1373 

-  65 

41 

-482 

-837 

—  1560 

—417 

—  477 

-  880 

-1558 

—339   —  267 

—  1026 

—  1562 

—242 

42 

—  44 

-  759 

—  1603 

-567 

-  89 

—  793 

—  1622 

—507   +  61 

—  920 

—  1635 

-4" 

43 

+  392 

-  655 

—  1524 

-689 

+  322 

-  685 

-1563 

-654 

+  408 

-  79i 

-1586 

—564 

44 

+  778 

—  530 

-1338 

-777 

+  715 

-  558 

-1385 

-769 

+  743 

-  643 

—  1428 

-696 

45 

+  1095 

-  388 

—  1067 

-828 

+  1054 

—  418 

—  1106 

—846   +1043 

-  484 

—  "59 

-800 

46 

+  1319 

—  240 

—  749 

-841 

+  1304 

-  268 

—  762 

-880 

+  1279 

—  319 

—  819 

-869 

47 

+  1445 

-  89 

—  392 

-815 

+  1453 

—  114 

-  384 

-872 

+  1429 

-  151 

—  427 

-896 

& 

+  1474 

+  57 

—  33 

-751 

+  J495 

+  33 

-   5 

—824   +1481 

+   12 

—   22 

-876 

49 

+  1405 

+  194 

+  292 

-651 

+  1438 

+  i/o 

+  338 

-740   +1430 

+  104 

+  349 

-813 

5<> 

+  1256 

+  3" 

+  56l 

-516 

+  1297 

+  292 

+  618 

—620  !  +1288 

+  296 

+  661 

-706 

51 

+  1048 

+  398 

+  741 

—354 

+  1085 

+  392 

+  829 

-467   +1078 

+  404 

+  886 

-565 

y- 

+  820 

+  446 

+  825 

—  177 

+  836 

+  468 

+  930 

—286   +  829 

+  484 

-j-IOII 

-395 

53 

+  609 

+  452 

+  819 

+  4 

+  575 

+  5ii 

+  932 

-  87   +  564 

+  534 

+  1031 

-205 

54 

+  432 

+  414 

+  734 

+  177 

+  342 

+  Sii 

+  839 

+116   +  313 

+  551 

+  955 

-  3 

55 

+  308 

+  342 

+  600 

+330 

+  159 

+  468 

+  672 

+313   +  97 

+  534 

+  790 

+206 

56 

+  237 

+  245 

+  451 

+463 

+  5i 

+  385 

+  465 

+490   -  55 

+  479 

+  556 

+4" 

57 

+  219 

+  127 

+  292 

+569 

+  26 

+  264 

+  246 

+629   —  125 

+  384 

+  297 

+597 

58 

+  247 

-   6 

+  161 

+640 

+  66 

+  "9 

+   72 

+726   —  105 

+  249 

+  66 

+742 

59 

+  312 

—  148 

+  89 

+665 

+  151 

-  38 

-  26 

+768   -  15 

+  87 

-  89 

+827 

COEFFICIENTS   FOR  DIRECT  ACTION. 


TABLE  III.— Continued. 
MUTUAL  PERIODIC  PERTURBATIONS  OF  VENUS  AND  THE  EARTH. 

The  term  of  long  period  is  omitted.     The  tabular  unit  is  o".oi  in  fu  and  6v',  and  10— 8  in  Sp  and  Sp'. 


i 

System  6. 

System  7. 

System  8. 

du 

8v' 

*t 

iff 

9* 

3v' 

if 

dp' 

du 

3v' 

if 

ip* 

0 

—  57 

+  45 

—  130 

+838 

—  225 

+  170 

—  33 

+759 

—  422 

+  3i9 

+  73 

+655 

i 

+  73 

—  129 

—  94 

+826 

—  no 

+   5 

—  39 

+773 

-  316 

+  156 

+  59 

+682 

2 

+  166 

—  292 

+  58 

+74i 

—   12 

-  155   +  56 

+726 

-  226 

—   3 

+  134 

+654 

3 

+  186 

—  426 

+  296 

+596 

+  38 

—  301   +  231 

+629 

—  177 

—  148 

+  272 

+58l 

4 

+  113 

-  519 

+  558 

+406 

+   21 

—  422   +  446 

+487 

—  179 

-  273 

+  434 

+471 

5 

-  55 

-565 

+  794 

+195 

—   82 

—  509   +663 

+312 

-  238 

-  375 

+  596 

+334 

6 

—  294 

-  565 

+  96i 

-  23 

-  266 

—  552   +  840 

+  "5   —  353   —  449 

+  733 

+  178 

7 

—  573 

-  526 

+  1039 

—232 

—  5" 

—  547   +  943 

—  94   —  522 

—  490 

+  827 

+  5 

8 

-  860 

-  453 

+  1017 

—421 

—  790 

—  493  I  +  950 

-298 

—  735 

—  490 

+  845 

—  175 

9 

—  1128 

—  351 

+  892 

-588 

-1066 

—  400 

+  855 

-483 

—  973 

—  442 

+  776 

—349 

10 

-1349 

—  228 

+  668 

-723 

—  1302 

—  275 

+  651 

-639 

—  I2OI 

-  348 

+  605 

-518 

ii 

—  1496 

-  88 

+  36i 

-823 

—  1470 

—  131 

+  368 

-758 

-1378 

—  217 

+  343 

-659 

12 

—  1549 

+  64 

—   9 

-879 

—  1549 

+  23 

+  23 

-836 

—1480 

-  62 

+  13 

-762 

13 

—  1404 

+  223 

—  405 

-893 

—  1524 

+  182 

—  352 

-876 

—1481 

+  107 

-  350 

-828 

14 

—  1335 

+  38i 

-  790 

-860 

—  1390 

+  339 

-  732 

-875 

-1378 

+  277 

—  7U 

-850 

IS 

—  1089 

+  533 

-1125 

-788 

—1150 

+  489 

—1076 

-834 

—1178 

+  440 

—  1048 

-831 

16 

-  775 

+  671 

-1382 

-682 

-815 

+  624 

-1355 

-757 

-875 

+  588 

—  1327 

—774 

17 

—  427 

+  796 

-1541 

-547 

—  419 

+  742 

—  1534 

—  640 

-  500 

+  7i8 

—  1521 

-680 

18 

—  73 

+  902 

—1592 

-396 

+    2 

+  833 

—  1595 

—496 

—  75 

+  821 

-1606 

—555 

19 

+  266 

+  087 

-1528 

-231 

+  409 

+  90i 

-1538 

-328 

+  370 

+  895 

-1567 

—406 

20 

+  563 

+  1046 

-1357 

—  59 

+  759 

+  944 

—  1364 

—151 

+  782 

+  934 

—  1402 

-238 

21 

+  799 

+  1073 

-1083 

+  114 

+  1030 

+  966 

—  1098 

+  29 

+  "24 

+  941 

—1126 

-  58 

22 

+  957 

+  1062 

-  731 

+277 

+  1205 

+  967 

-  754 

+203 

+  1362 

+  920 

-  766 

+  123 

2.3 

+  1023 

+  1009 

—  349 

+425 

+  1274 

+  943 

—  359 

+366 

+  1479 

+  876 

-  356 

+298 

24 

+  1001 

+  919 

+  47 

+544 

+  1232 

+  890 

+  4O 

+512 

+  1467 

+  812 

+  70 

+459 

25 

+  899 

+  801 

+  412 

+632 

+  1087 

+  809 

+  461 

+634 

+  1333 

+  735 

+  486 

+599 

26 

+  737 

+  663 

+  726 

+687 

+  842 

+  696 

+  816 

+726 

+  1083 

+  641 

+  861 

+713 

27 

+  540 

+  5'7 

+  962 

+7l6 

+  541 

+  565 

+  1000 

+784 

+  742 

+  531 

+  "65 

+795 

28 

+  3l8 

+  3«5 

+  "42 

+729 

+  207 

+  419 

+  1266 

+809 

+  340 

+  406 

+1368 

+846 

29 

+  CO 

+  213 

+  1251 

+728 

—  121 

+  270 

+J339 

+802 

-  91 

+  268 

+1457 

+861 

30 

—  139 

+  57 

+  1282 

+721 

—  422 

+  121 

+1321 

+772 

-  498 

+  126 

+1428 

+843 

3i 

-  359 

—  IO2 

+  1236 

+705 

-  677 

—   23 

+1230 

+727 

-  849 

—  n 

+  1294 

+796 

32 

-  562 

—  262 

+  1118 

+679 

-884 

—  164 

+1071 

+673 

—  1126 

—  140 

+  1084 

+726 

33 

—  73i 

—  422 

+  926 

+645 

—  1037 

-  306 

+  855 

+615 

—1312 

—  260 

+  813 

+642 

34 

-852 

—  574 

+  675 

+599 

—  1128 

—  445 

+  589 

+553 

—  1406 

-  370 

+  Sio 

+548 

35 

—  923 

-  718 

+  374 

+545 

—"53 

—  579 

+  284 

+487 

—1418 

—  474 

+  187 

+453 

36 

-  935 

-852 

+  39 

+480 

—  1108 

—  709 

—  40 

+414 

—1346 

—  577 

-  136 

+357 

37 

-  886 

—  972 

—  315 

+399   —  991 

-  828 

—  365 

+334 

—  I2O2 

-  676 

—  449 

+265 

38 

-  773 

—  1071 

-683 

+300   —  803 

—  933 

-  688 

+246 

-982 

-  767 

—  743 

+  177 

39 

-  589 

—i  143  ;  —1024 

+  178   -  558 

—  1024 

-983 

+  150 

—  697 

-850 

—  997 

+  88 

40 

—  335 

—1177   —1317 

+  33   -  262 

—  1093 

—  1240 

+  45 

—  362 

—  920 

—1203 

—  3 

41 

-  28 

—1165  j  —1526 

—  130   +  79 

—"39 

-1436 

—  72 

+   12 

—  977 

—  1354 

-96 

42 

+  313 

—  noo   —  1622 

—297   +  450 

—  "47 

—1550 

—201 

+  407 

—  1017 

-1438 

-192 

43 

+  650 

—  989 

—  1599 

-458 

+  825 

—  III2 

—1556 

—337 

+  80S 

—1036 

-1454 

-291 

44 

+  959 

-  837 

—  1456 

-600 

+  "72 

—1024 

—1451 

-473 

+"90 

—  1024 

-1381 

—393 

45 

+  1215 

—  658   —1207 

-717   +1462 

-887 

-1233 

-599 

+1535 

—  974 

—1213 

-491 

46 

+  1400 

-  466   -  83  1 

—  80  1   +1660 

—  709 

—  930 

—703 

+1805 

-  878 

-  961 

-581 

47 

+  1505 

—  269   —  496 

—850   +1753 

—  502 

—  564 

—773 

+  1974 

-  737 

—  629 

-655 

48 

+  1524 

-  77   -  87 

-863  •  +1738 

-  285 

-  167 

—807 

+2021 

-  555 

-  248 

-70S 

49 

+  1454 

+  106   +  307 

-837 

+1629 

-  67 

+  222 

—803 

+  1941 

—  342 

+  135 

—725 

50 

+  1298 

+  270 

+  651 

-769 

+1433 

+  137 

+  576 

-758 

+  1741 

—  125 

+  488 

-706 

Si 

+  1070 

+  407 

+  909 

-653 

+  1168 

+  319 

+  866 

-675 

+  1450 

+  9i 

+  785 

—650 

52 

+  802 

+  5i3 

+  1062 

—500 

+  861 

+  471 

+  1063 

-556 

+  1100 

+  287 

+  998 

-555 

53 

+  52i 

+  582 

+  IIOI 

-316 

+  534 

+  586 

+"5i 

—403 

+  720 

+  454 

+  1118 

—425 

54 

+  256 

+  610 

+  1045 

—  117 

+  225 

+  654 

+  "29 

—225 

+  346 

+  58o 

+"38 

-271 

55 

+  34 

+  599 

+  890 

+  93 

—  43 

+  677 

+  988 

-  27 

+   7 

+  662 

+  1051 

—  97 

56 

-  135 

+  553   +  671 

+299 

—  239 

+  651 

+  773 

+  180 

-  266 

+  693 

+  875 

+  90 

57 

—  224 

+  472   +  414 

+49i 

—  348 

+  579 

+  520 

+379 

—  446 

+  670 

+  639 

+273 

58 

—  237 

+  356   +  163 

+656 

—  371 

+  467 

+  272 

+552 

-  522 

+  592 

+  392 

+440 

59 

-  174 

+  212   :  —   33 

+776 

—  322 

+  328 

+  76 

+683 

—  504 

+  471 

+  188 

+572 

ACTION   OF  THE   PLANETS  ON   THE  MOON. 


TABLE    III.  —Concluded. 
MUTUAL  PERIODIC  PERTURBATIONS  OF  VENUS  AND  THE  EARTH. 

The  term  of  long  period  is  omitted.     The  tabular  unit  is  o".oi  in  tu  and  dv',  and  to— 8  in  <$/>  and 


System  9. 

Systen/io. 

System  II. 

i 

du 

dv' 

dp 

9ff 

du 

Sv' 

*P 

9p> 

du 

ilp' 

df> 

dp' 

0 

-  598 

+447 

+  196 

+563 

-  548 

+407 

+  300 

+495 

—  205 

+  158 

+  341 

+471 

i 

—  544 

+3i8 

+  174 

+583 

-  616 

+355 

+  295 

+506 

-  371 

+  169 

+  372 

+454 

2 

—  482 

+176  ;  +  237 

+552 

—  660 

+285 

+  355 

+469 

-  540 

+  175 

+  468 

+393 

3 

—  444 

+  36   +  357 

+487 

-698 

+200 

+  465 

+398 

-  718 

+  179 

+  591 

+306 

4 

—  446 

—  90 

+  493 

+394 

—  747 

+  109 

+  579 

+301 

-897 

+  176 

+  700 

+207 

5 

—  493 

-196 

+  613 

+285 

-  811 

+  19 

+  673 

+  193 

-1066 

+  165 

+  767 

+  103 

6 

—  576 

-278 

+  699 

+  162 

-889 

-  62 

+  727 

+  82 

—  1208 

+  141 

+  783 

+  o 

7 

-  690 

-338 

+  744 

+  26 

—  980 

—127 

+  732 

—  32 

—1321 

+  109 

+  757 

—  104 

8 

-  828 

—370 

+  736 

—  121 

—1073 

-171 

+  681 

—142 

—1409 

+  77 

+  662 

—205 

9 

-981 

-371 

+  667 

—272 

—1164 

—  194 

+  577 

-254 

-1470 

+  51 

+  5i6 

—300 

10 

-1138 

—335 

+  522 

—42O 

—1249 

—193 

+  416 

—364 

—1497 

+  39 

+  342 

-384 

II 

—  1274 

—260 

+  298 

-553 

—1290 

-166 

+  203 

-472 

-1484 

+  41 

+  "4 

—456 

12 

—  1360 

-147 

__     ny 

-661 

—1310 

—113 

—  60 

-569 

-1423 

+  59 

—  147 

-517 

13 

—  1372 

—  i 

—  351 

—740 

—1274 

-  29 

-  304 

—648 

—  1314 

+  95 

—  427 

-566 

M 

-1288 

+  164 

—  715 

-783 

—1176 

+  83 

—  693 

—699 

—"54 

+  147 

-  7" 

—603 

IS 

—  1106 

+339 

—1055 

-786 

—  1000 

+222 

—  1015 

-716 

-  938 

+220 

-987 

—624 

16 

-  835 

+504 

-1328 

—753 

-  741 

+374 

—  1301 

-695 

-  669 

+3H 

-1237 

—620 

17 

—  493 

+654 

—1520 

-682 

—  415 

+530 

—  15" 

-637 

—  346 

+420   -1438 

-583 

18 

-  106 

+779 

—  1609 

—576 

—  40 

+671 

—  1617 

-549 

+  19 

+536   -1561 

—510 

19 

+  308 

+876 

-1582 

—440 

+  353 

+791 

—  1605 

-431 

+  413 

+651 

—  1579 

—403 

20 

+  7io 

+938 

-1438 

-282 

+  726 

+878 

-1469 

—293   +  793 

+749 

—  1479 

—270 

21 

+  1072 

—  1180 

-108 

+  1049 

+933 

—  1225 

—  135   +1126 

+821 

—  1260 

-116 

22 

+  1359 

+948 

—  820 

+  68 

+  1323 

+949 

-  880 

+  33 

+  1387 

+862 

—  940 

+  40 

23 

+  1532 

+899 

—  399 

+242 

+  1501 

+927 

—  467 

+207 

+  1547 

+867 

-  544 

+  202 

24 

+  1580 

+816 

+  54 

+405 

+  1572 

+863 

—  13 

+376 

+  1600 

+835 

—  IOI 

+359 

25 

+  1493 

+7U 

+  494 

+551 

+  1525 

+765 

+  446 

+530 

+1546 

+769 

+  355 

+509 

26 

+  1278 

+592 

+  885 

+677 

+  1353 

+632 

+  872 

+659   +1379 

+667 

+  79i 

+644 

27 

+  957 

+466 

+  1210 

+778 

+  1070 

+479 

+  1221 

+759   +1116 

+533 

+  U74 

+757 

28 

+  558 

+336 

+  1436 

+849 

+  701 

+3i6 

+  M7I 

+833   +  766  >  +372 

+  1460 

+841 

29 

+  106 

+205 

+  1545 

+884 

+  271 

+  153 

+  1600 

+874   +  353 

+  192 

+  1627 

+887 

30 

—  355 

+  73 

+  1534 

+883 

-  179 

—  I 

+  1609 

+886   -  84 

+  6 

+  1659 

+896 

31 

-  79i 

-  59 

+  1404 

+848 

-  621 

—  142 

+  1501 

+867   -  515 

-171 

+  1565 

+870 

32 

—  1164 

-183 

+  1169 

+780 

—  1026 

-270 

+  I278 

+815 

-  909 

-330 

+  1357 

+814 

33 

—1439 

—297 

+  861 

+690 

-1358 

-383 

+  96l 

+733 

—1234 

-466 

+  1050 

+735 

34 

—  1594 

—393 

+  497 

+58l 

—  1592 

-477 

+  574 

+621 

—1476 

-573 

+  673 

+632 

35 

—  1633 

-471 

+  134 

+465 

—  1704 

-550 

+  154 

+491 

—1629 

—652 

+  250 

+Sn 

36 

—  1560 

-537 

-  225 

+345 

-1684 

—602 

+351 

-1661 

-707 

-  183 

+370 

37 

—  1402 

-594 

+226 

—  1549 

—630 

—  620 

+209 

—1579 

-733 

-  596 

+216 

38 

-1173 

-045 

—  818 

+  H5 

-1317 

—639 

—  923 

+  77   -1384 

-730 

—  959 

+  60 

39 

-  879 

—694 

-1058 

+  n 

—  1017 

-637 

—  1149 

—  43 

—1098 

—703 

—  1232 

-  So 

40 

-  540 

-740 

-1237 

-84 

—  674 

-631 

—  1301 

-ISO 

-  752 

—654 

—  1403 

—219 

4i 

-  159 

-783 

-1347 

—  169 

—  303 

-627 

-244 

-  37S 

—595 

—  1473 

-324 

42 

+  244 

-813 

—1390 

-245 

+  80 

-626 

-324 

+   3 

—532 

—  1450 

-403 

43 

+  651 

-838 

—1365 

—315 

+  465 

-626 

—  1342 

—391  ,  +  364   —477 

—  1357 

—462 

44 

+  1039 

-849 

-1275 

-383 

+  835 

—623 

—1225 

—442   +  697   —432 

—  1208 

—504 

45 

+  1399 

-845 

—  II2I 

—449 

+  1181 

-617 

—  1049 

—478   +1001   —306 

—  1009 

—531 

46 

+  1708 

—821 

—  911 

—509 

+  1482 

-607 

-  832 

—502   +1264   —368 

-  778 

-542 

4? 

+  1946 

-768 

—  642 

—562 

+  1728 

-591 

-  580 

—517   +1480  ;  —346 

-  520 

-536 

48 

+2089 

-681 

-  322 

—60  1 

+  1908 

-566 

-  303 

—526   +1640   —328 

—  248 

—515 

49 

+2122 

-S<S 

+  27 

—621 

+2014 

-524 

—  IS 

—527   +1743   —312 

+  17 

-481 

50 

+2026 

—  39J 

+  372 

-611 

+2027 

—460 

+  276 

-514 

+  1784   -298 

+  265 

—434 

Si 

+  1803 

-207 

+  678 

-570 

+  1939 

—371 

+  548 

—479 

+  1761  i  —283 

+  487 

-378 

52 

+  1482 

—  II 

+  912 

—  4C9 

+  1744 

-254  i  +  779 

—418   +1679   —259 

+  673 

—314 

53 

+  1093 

+  178 

+  1054 

-396 

+  1454 

—  116  i  +  942 

—327    +1528    —221 

+  813 

—236 

54 

+  679 

+344 

+  1097 

—269 

+  1097 

+  27 

+  1017 

—213    +1313    —l69 

+  895 

—145 

55 

+  279 

+477 

+  1042 

-117 

+  709 

+  166 

+  995 

—  8O   :  +IO44   j  —  IOI 

+  900 

—  31 

56 

—  71 

+569 

+  006 

+  49 

+  331 

+284 

+  884 

+  66   +  743   —  27 

+  825 

+  IOO 

57 

—  344 

+612 

+  7io 

+217 

+   3 

+371 

-  716 

+209   +  451   +  43 

+  686 

+234 

58 

-  523 

+603 

+  497 

+370 

—  254 

+418 

+  536 

+337 

+  189+99 

+  525 

+354 

59 

-  601 

+548 

+  3t2 

+489 

—  434 

+430 

+  38S 

+437 

-  27 

+  137 

+  397 

+437 

COEFFICIENTS   FOR  DIRECT  ACTION. 


53 


TABLE   IV«. 
PERTURBATIONS  OF  THE  G-COORDINATE  X  OF  VENUS. 

The  tabular  unit  is  io-8. 


Sys- 
tern 
i 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

0 

+  47 

+  57 

+  92 

+  137 

+  181 

+217 

+218 

+  184 

+142 

+107 

+  71 

+  52 

i 

+  37 

+  50 

+  81 

+  118 

+  159 

+197 

+212 

+  183 

+140 

+  92 

+  52 

+  30 

2 

—  ii 

+  ii 

+  41 

+  7i 

+  109 

+149 

+  176 

+  158 

+H4 

+  59 

+  8 

—  23 

3 

—  92 

—  59 

—  24 

+  5 

+  39 

+  75 

+  112 

+  iii 

+  72 

+  9 

—  52 

—  97 

4 

—  193 

—154 

—  in 

—  79 

—  50 

—  14 

+  26 

+  46 

+  14 

-48 

—  122 

-183 

5 

—305 

-269 

—217 

-183 

-156 

—  122 

—  79 

-  41 

—  55 

—in 

-191 

-268 

6 

—412 

-396 

-338 

—290 

-267 

—237 

—197 

—  146 

-136 

-179 

-26l 

-348 

7 

—506 

-519 

—465 

—405 

-380 

-354 

-318 

-263 

—230 

-251 

-326 

—421 

8 

-581 

-628 

-586 

—519 

-487 

-468 

—437 

-384 

—331 

—330 

-386 

—482 

9 

-636 

—715 

-696 

—629 

-583 

-569 

-546 

-502 

-437 

—410 

—443 

—530 

IO 

-668 

—763 

-780 

-718 

-661 

—650 

-636 

-599 

—539 

-488 

—494 

-565 

ii 

—674 

—777 

-827 

-783 

-718 

-699 

—697 

—672 

—619 

-556 

-534 

-582 

12 

-660 

—759 

-828 

—814 

—745 

—714 

-722 

—710 

-669 

—606 

-561 

-580 

13 

—621 

—710 

—790 

-803 

-745 

—699 

-707 

-712 

-682 

-625 

—569 

-560 

14 

-559 

-639 

—713 

—749 

—709 

-652 

—652 

-671 

—659 

—  610 

—552 

-523 

IS 

—479 

-541 

—614 

-659 

—640 

-580 

—564 

-591 

-596 

—559 

—507 

-466 

16 

-383 

—431 

—495 

-542 

-540 

-487 

—452 

-475 

—496 

—476 

—432 

-389 

17 

—279 

-308 

—364 

—410 

—417 

-374 

-325 

—337 

-368 

-366 

—331 

-298 

18 

—  171 

-183 

-231 

-273 

-283 

-251 

-195 

-187 

—221 

-236 

-213 

-187 

19 

-  54 

-  61 

-98 

—  137 

—  149 

—123 

-  68 

-38 

—  66 

-  95 

-  84 

-  64 

20 

+  60 

+  56 

+  25 

—  13 

-  26 

—  i 

+  48 

+  93 

+  80 

+  46 

+  41 

+  58 

21 

+  170 

+160 

+  136 

+  IOO 

+  82 

+  106 

+151 

+203 

+209 

+179 

+155 

+  171 

22 

+269 

+253 

+229 

+197 

+  174 

+192 

+236 

+288 

+312 

+291 

+262 

+266 

23 

+350 

+330 

+305 

+275 

+249 

+259 

+299 

+350 

+384 

+377 

+348 

+342 

24 

+406 

+392 

+363 

+333 

+307 

+306 

+341 

+387 

+430 

+434 

+412 

+398 

25 

+441 

+436 

+407 

+373 

+347 

+338 

+367 

+412 

+451 

+466 

+456 

+437 

26 

+461 

+460 

+435 

+400 

+374 

+36i 

+378 

+418 

+457 

+475 

+476 

+464 

27 

+468 

+47i 

+451 

+417 

+388 

+372 

+380 

+418 

+454 

+478 

+483 

+478 

28 

+476 

+475 

+459 

+426 

+394 

+381 

+379 

+411 

+447 

+472 

+481 

+484 

29 

+480 

+475 

+463 

+432 

+397 

+381 

+379 

+405 

+440 

+465 

+475 

+485 

30 

+484 

+479 

+467 

+438 

+401 

+38o 

+380 

+401 

+434 

+459 

+473 

+479 

31 

+400 

+484 

+476 

+445 

+406 

+380 

+38o 

+397 

+431 

+457 

+470 

+476 

32 

+491 

+492 

+483 

+455 

+414 

+380 

+379 

+392 

+426 

+456 

+469 

+473 

33 

+485 

+497 

+488 

+460 

+417 

+379 

+375 

+387 

+419 

+451 

+467 

+466 

34 

+468 

+492 

+489 

+463 

+420 

+377 

+362 

+375 

+403 

+437 

+456 

+456 

35 

+439 

+468 

+479 

+458 

+414 

+366 

+341 

+354 

+377 

+411 

+432 

+436 

36 

+397 

+426 

+448 

+437 

+392 

+342 

+3io 

+314 

+338 

+368 

+391 

+399 

37 

+340 

+360 

+394 

+396 

+353 

+301 

+262 

+260 

+282 

+3ii 

+331 

+343 

38 

+263 

+270 

+309 

+327 

+293 

+240 

+  196 

+  184 

+208 

+236 

+255 

+264 

39 

+  167 

+  166 

+  199 

+230 

+211 

+  155 

+  109 

+  90 

+  112 

+  141 

+  161 

+  166 

40 

+  56 

+  47 

+  69 

+  107 

+  104 

+  Si 

+   2 

—  20 

—  4 

+  33 

+  55 

+  57 

41 

-  67 

-  80 

—  74 

-  38 

—  22 

-67 

—  121 

-145 

-127 

—  90 

—  59 

-  58 

42 

-188 

—206 

—215 

—  188 

-161 

—192 

—254 

-283 

—265 

—222 

-180 

-172 

43 

—301 

—325 

—344 

-336 

-306 

—319 

-381 

—420 

—407 

-357 

-304 

-282 

44 

-397 

—430 

—459 

—464 

—443 

-438 

—494 

—549 

-543 

-486 

-423 

-385 

45 

—473 

—510 

—545 

-568 

—557 

—544 

-589 

-656 

-665 

—609 

—534 

-475 

46 

-528 

-56o 

-605 

—634 

-638 

-626 

-655 

—729 

-760 

-712 

-628 

-555 

47 

-566 

-578 

-626 

-665 

-681 

-678 

—694 

-764 

-818 

-791 

—  702 

-617 

48 

-585 

—574 

-611 

-660 

-685 

—692 

—703 

—759 

-833 

-833 

—749 

-657 

49 

-588 

—547 

-566 

—619 

-653 

-668 

-679 

—723 

-805 

-838 

-775 

-675 

50 

—572 

-508 

-505 

-549 

-588 

—609 

-627 

-659 

-737 

-796 

-766 

-668 

Si 

—539 

—457 

-431 

—454 

-499 

-525 

-545 

—571 

—639 

-716 

-723 

—639 

52 

—491 

—401 

—347 

-350 

-391 

—423 

-447 

—470 

-525 

-605 

-644 

-589 

53 

—431 

—339 

-268 

—248 

—276 

—3n 

—333 

—357 

—  401 

-477 

-535 

-518 

54 

-362 

-274 

—  193 

-152 

—  162 

—  197 

—222 

—247 

-281 

-346 

—414 

—428 

55 

-283 

-205 

—  122 

-69 

-  58 

-88 

—  III 

—137 

-167 

—219 

-286 

-324 

56 

-109 

—  134 

-  58 

+   2 

+  31 

+  14 

—  14 

—  37 

-64 

-106 

—164 

—211 

57 

—  114 

-  67 

+  60 

+  IOI 

+  102 

+  73 

+  48 

+  20 

-  14 

-  60 

-105 

58 

-  38 

-  8 

+  44 

+  104 

+151 

+  167 

+  144 

+  H5 

+  85 

+  54 

+  16 

-  18 

59 

+  19 

+  36 

+  78 

+130 

+179 

+208 

+  194 

+  161 

+  128 

+  94 

+  59 

+  36 

54 


ACTION   OF  THE   PLANETS  ON   THE   MOON. 


TABLE    IV*. 
PERTURBATIONS  OF  THE  G-COORDINATE  Y  OF  VENUS. 

The  tabular  unit  is  10— 8. 


Sys- 
tem 
i 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

o 

—164 

—370 

—379 

—274 

-176 

-  78 

+  42 

+  160 

+300 

+423 

+385 

+  146 

i 

—  100 

—357 

—436 

-365 

-275 

-199 

-  87 

+  39 

+181 

+337 

+377 

+202 

2 

—   22 

—310 

—454 

—430 

-350 

-290 

—203 

-  77 

+  68 

+239 

+348 

+251 

3 

+  70 

—224 

—424 

-458 

—394 

-345 

-287 

—  175       —  30 

+  145 

+304       +294 

4 

+172 

—ill 

—350 

-438 

—405 

-360 

—330 

—246 

-108 

+  65 

+252    ,    +324 

5 

+273 

+   22 

-236 

—373 

-378 

—341 

-328 

-281 

-162 

+     3 

+200       +339 

6 

+355 

+  164 

—  97 

-272 

-318 

-294 

-289 

—277       -190 

-  38 

+  154       +334 

7 

+410 

+298 

+  53 

—  148 

—231 

—228 

—225 

—239 

-193 

-  65 

+  ii5       +313 

8 

+435 

+406 

+202 

—  II 

—  124 

—  146 

—  144 

—  171 

-167 

-  72 

+  89       +286 

9 

+436 

+48l 

+340 

+  131 

-    6 

-  54 

-  59 

-  87       -114 

—  62 

+  73       +258 

10 

+415 

+523 

+452 

+267 

+  114 

+  46 

+  32 

+    7       -  38 

-  31 

+  70 

+232 

ii 

+389 

+531 

+533 

+390 

+230 

+  146 

+  120 

+  98       +49       +19 

+  78 

+214 

12 

+36l 

+519 

+583 

+490 

+339 

+244 

+205 

+  185    i    +139       +  86 

+  101 

+205 

13 

+339 

+495 

+602 

+571 

+442 

+338 

+200 

+267 

+227       +165 

+  142 

+207 

14 

+331 

+472 

+600 

+626 

+533 

+429    ,    +372 

+348 

+315       +254 

+201 

+229 

IS 

+339 

+457 

+590 

+660 

+610 

+515        +454 

+427 

+309        +348 

+279 

+  268 

16 

+363 

+457 

+579 

+673 

+673 

+597    i    +536 

+508 

+484        +440 

+371 

+329 

17 

+406 

+472 

+575 

+679 

+721 

+673 

+6l5 

+588 

+568        +534 

+473 

+409 

18 

+469 

+502 

+582 

+678 

+749 

+738 

+600 

+667 

+650       +624 

+579    ,    +505 

19 

+550 

+548 

+602 

+680 

+76o 

+787 

+757 

+735 

+729 

+710 

+675       +610 

20 

+639 

+606 

+629 

+685 

+759 

+812 

+809 

+792 

+793 

+786 

+76i 

+710 

21 

+729 

+668 

+660 

+692 

+747 

+814 

+837 

+832 

+837 

+845 

+829 

+795 

22 

+804 

+729 

+691 

+698 

+729 

+788 

+835 

+847 

+858 

+877 

+875 

+855 

23 

+848 

+779 

+715 

+606 

+704 

+743 

+799 

+830 

+846 

+875 

+888 

+880 

24 

+854 

+801 

+725 

+681 

+671 

+681        +729 

+778 

+801 

+833 

+861 

+864 

25 

+812 

+786 

+717 

+653 

+621 

+606        +630 

+683 

+7i7 

+752 

+792 

+809 

25 

+723 

+722 

+676 

+602 

+552 

+SI7        +510 

+551 

+594 

+632 

+676 

+7io 

27 

+594 

+609 

+593 

+53i 

+465 

+416        +383 

+397 

+441 

+478 

+522 

+567 

28 

+428 

+455 

+467 

+428 

+360 

+297       +2.48 

+228 

+262 

+301 

+34i 

+390 

29 

+237 

+269 

+301 

+295 

+239 

+174       +H3 

+  66 

+  70 

+  106 

+  138 

+  187 

30 

+  26 

+  67 

+  107 

+  132 

+  100 

+   40           —   22 

—  po 

-116 

—  92 

-  65 

-  27 

31 

-187 

—  142 

—099 

—  54 

-  Si 

-   98      1     -155 

-227       -280 

-283 

-261 

-235 

32 

-388 

—340 

—300 

—250 

—  216 

-237    !    -284 

—346       —416 

—449 

—440 

—424 

33 

-565 

—523 

-484 

—439 

-389 

-375    I    -403 

—451    i    —520 

-580 

-589 

-582 

34 

—702 

-673 

-637 

-602 

—547 

-505 

-509 

-538 

—594 

-667 

-702 

—703 

35 

-796 

-785 

-754 

-729 

-685 

—625 

-601 

—609 

—644 

-711 

-770 

-787 

36 

-846 

-852 

-835 

—814 

-789 

-728 

-679 

-664 

-675 

-723 

-788 

—829 

37 

-857 

-875 

-872 

-860 

-850 

-807 

—743 

—704 

—691 

—712 

—774 

-830 

38 

-832 

-857 

-869 

-867 

—869 

-850 

—791 

—729 

-695 

-688 

—729 

—794 

39 

-779 

—  810 

-833 

—840 

—850 

-858 

-820 

—749 

-690 

—659 

-669 

—729 

40 

—704 

-741 

-769 

-787 

—802 

-835 

-827 

-761 

-685 

—631 

-608 

—647 

41 

-613 

—659 

-690 

-713 

-736 

-778 

-810 

-767 

-682 

—607 

-554 

-561 

42 

-518 

-568 

-601 

—629 

-655 

—704        —766 

-763 

-680 

-589 

—512 

-483 

43 

—428 

-475 

—513 

-539 

-568 

—  618        —702 

—743 

-686 

-582 

-484 

—421 

44 

-350 

-383 

—  426 

-452 

-478 

—528      ;     —622 

—705 

-689 

-586 

-468 

—379 

45 

-289 

—299 

—341 

-369 

-393 

—435     '    —530 

-645 

-68  1 

-597 

-469 

—353 

46 

—249 

-226 

-260 

—291 

—310 

—344    !    —432 

—569 

-659 

—  612 

-480 

—345 

47 

—227 

-168 

-183 

—214 

—229 

—254        —329 

—472 

—  610 

-618 

—499 

—352 

48 

—220 

-126 

—  in 

-139 

-152 

—  166    ;    —223 

—360 

—532 

-606 

—521 

-363 

49 

—227 

—103 

—  47 

-  64 

-  77 

-  79 

-116 

-235 

—425 

-566 

-535 

-387 

50 

—240 

-  94 

—    3 

+     9 

—      2 

+    8 

—    10 

—  104 

-295 

-488 

-530 

—410 

51 

-258 

-  96 

+  25 

+  75 

+  74 

+  88 

+  97 

+  30 

—  148 

—377 

—497 

—424 

52 

-278 

-105 

+  40 

+  124 

+  143 

+  163 

+  192 

+  160 

+    8 

—235 

—427 

—427 

S3 

—297 

-124 

+  35 

+  151 

+200 

+226 

+270 

+276 

+  161 

-  77 

—324 

—407 

54 

—314 

-150 

+  13 

+  I5I 

+234 

+272 

+327 

+368 

+299 

+  84 

—  194 

-359 

55 

—319 

—  184 

—  21 

+  124 

+237 

+298 

+358 

+428 

+409 

+235 

-  49 

-285 

56 

—311 

—226 

—  68 

+  74 

+205 

+291 

+360 

+449 

+483 

+361 

+  92 

—  192 

57 

—2pO 

-273 

-132 

+    8       +136 

+247 

+329 

+427 

+509 

+4-18 

+215 

—  94 

58 

—257 

-319 

—212 

„__„     •*•* 

+  42 

+  164 

+264 

+365 

+481 

+487 

+305 

+     I 

59 

-215 

-355 

-298 

-173 

-  66 

+  50 

+  165 

+274 

+407 

+477 

+36l 

+  81 

COEFFICIENTS   FOR  DIRECT  ACTION. 


55 


TABLE   V. 

RECTANGULAR  G-COORDINATES  X  AND  Y  OF  VENUS,  REFERRED,  IN  EACH 
SYSTEM,  TO  AN  Axis  OF  X  PASSING  THROUGH  THE  MEAN  SUN. 


i 

System  o. 

System  i. 

System  2. 

X 

r 

X 

r 

X 

r 

0 

+0.264  4388 

+0.005  3159 

+0.267  8642 

+0.0172810 

+0.273  5900 

+0.024  1368 

i 

+0.2682411 

—0.070  6564 

+0.271  9461 

—0.058  8025 

+0.277  8772 

—0.0518927 

2 

+0.280  0227 

-0.1458383 

+0.2840031 

—0.1340369 

+0.290  1029 

—0.1270162 

3 

+0.299  6585 

—0.219  3893 

+0.303  8978 

—0.207  5810 

+0.310  1265 

—0.200  3988 

4 

+0.326  9346 

—0.290  4863 

+0.331  4038 

—0.278  6154 

+0.337  7133 

—0.271  2285 

5 

+0.361  5487 

—0.358  3344 

+0.366  2082 

—0.346  3496 

+0.372  5452 

—0.338  7248 

6 

+0.403  1159 

—0.422  1763 

+0.4079159 

—0.4100328 

+0.414  2234 

—  O.4O2  1483 

7 

+0.451  1712 

—0.481  2993 

+0.456  0557 

—0.468  9619 

+0.462  2754 

—0.460  8077 

8 

+0.505  1754 

—0.535  0445 

+0.5100841 

—0.522  4898 

+0.5161594 

—0.5I4067I 

9 

+0.564  5219 

—0.5828139 

+0.569  3924 

—0.570  0313 

+0.575  2710 

—0.56l  3528 

10 

+0.628  5438 

—0.624  0792 

+0.633  3160 

—  0.61  1  0692 

+0.6389518 

—O.6O2  l6O2 

ii 

+0.696  5229 

-0.658  3852 

+0.701  1382 

—0.645  1617 

+0.7064955 

—0.636  0564 

12 

+0.767  6965 

—0.685  3576 

+0.772  1023 

—0.671  9440 

+0.777  1566 

-0.662  6856 

13 

+0.841  2692 

—0.704  7061 

+0.845  4198 

—0.691  1353 

+0.850  1572 

—0.68  1  7730 

14 

+0.9164199 

—0.716  2270 

+0.920  2784 

—0.702  5389 

+0.924  6978 

—0.693  1251 

15 

+0.9923125 

—0.7198055 

+0.995  8539 

—  0.706  0456 

+0.9999648 

—0.696  6325 

16 

+  1.068  1057 

—0.7154169 

+  1.0713157 

—0.701  6335 

+  1.075  1404 

—0.692  2716 

17 

+  1.1429619 

-0.703  1247 

+  1.1458394 

-0.6893674 

+  1.1494106 

—0.680  1025 

18 

+  1.2160577 

-0.683  0801 

+  1.2186136 

—0.669  3977 

+  1.2219735 

—0.6602691 

19 

+  1.2865933 

—0.655  5200 

+  1.2888493 

-0.641  9577 

+  1.2920488 

—0.632  9963 

20 

+  I-353799I 

—0.620  7640 

+  1.3557887 

—0.607  3599 

+  1.3588839 

-0.5985883 

21 

+  1.4169464 

—  0.579  2086 

+  1.4187118 

—0.565  0943 

+  1.421  7633 

—0.557  4240 

22 

+  1.4753529 

—0.531  3238 

+  1.4769448 

—0.5183206 

+  14800138 

—0.5099523 

23 

+  1.5283911 

—0.477  6470 

+  1.5298660 

—04648651 

+  I.5330I33 

—04566884 

24 

+  1-5754942 

—0.418  7748 

+  1.5769123 

—  0.4062148 

+  1.580  1954 

—0.398  2084 

25 

+  1.6161628 

-0.355  3577 

+  1.6175843 

—0.343  0090 

+  1.6210558 

—0.335  1418 

26 

+  1.6499691 

—0.2880919 

+  1.6514516 

—0.275  9346 

+  1.655  1564 

—0.268  1673 

27 

+  I.6765593 

—0.2177116 

+  1.678  1568 

—0.205  7162 

+  1.682  1302 

—0.1980049 

28 

+  1.6956592 

—0.1449820 

+  1.6974188 

—0.133  1099 

+  1.7016857 

—0.1254065 

29 

+  1.7070734 

—0.070  6903 

+  1.7090351 

—0.058  8960 

+  1.7136087 

—0.051  1511 

30 

+  1.7106804 

+0.0043615 

+  1.7128842 

+0.016  1297 

+  1.7177659 

+0.023  9652 

3i 

+  1.7064775 

+0.079  3664 

+  1.7089249 

+0.091  1628 

+  1.714  1050 

+0.099  1360 

32 

+  1.6944891 

+O.I535I90 

+  1.6971990 

+0.165  3988 

+1.702  6556 

+0.1735529 

33 

+  1.6748587 

+0.226  0244 

+  1.6778288 

+0.238  0408 

+  1.6835308 

+0.2464124 

34 

+  1.6478008 

+0.296  1060 

+  1.6510177 

+0.308  3085 

+  1.6569253 

+0.3169261 

35 

+  1.6136080 

+0.3630127 

+  1.6170473 

+0-375  4452 

+  1.623  1144 

+0.384  3269 

36 

+  1.5726476 

+0.426  0265 

+  1.5762761 

+0.4387253 

+  1.5824506 

+0.4478781 

37 

+  1-5253583 

+0.484  4694 

+  1.529  1350 

+0497  4622 

+  1.5353622 

+0.506  8825 

38 

+  1.4722450 

+0.537  7122 

+  1.4761240 

+0.5510155 

+  1.4823484 

+0.560  6879 

39 

+  1.4138753 

+0.585  1789 

+  1.4178080 

+0.598  7973 

+  1.4239759 

+0.608  6959 

40 

+  1.3508731 

+0.626  3539 

+  1.3548096 

+0.640  2802 

+  1.3608720 

+0.650  3687 

41 

+  1.2839122 

+0.660  7876 

+  1.2878040 

+O.675  OO2O 

+  1.2937184 

+0.685  2350 

42 

+  1.2137108 

+0.6880995 

+  1.2175124 

+0.702  5721 

+  1.2232454 

+0.7128968 

43 

+  1.1410227 

+0.707  9845 

+  1.1446940 

+0.722  6753 

+  1.1502220 

+0.7330331 

44 

+  1.0666304 

+0.7202157 

+  1.070  1388 

+0.735  0768 

+  1.0754484 

+0.745  4066 

45 

+0.991  3369 

+0.724  6478 

+0.994  6591 

+0.739  6239 

+0.999  7488 

+0.749  8650 

46 

+0.9I5957I 

+0.721  2182 

+0.919  0806 

+0.7362498 

+0.923  9587 

+0.746  3434 

47 

+0.841  3093 

+0.709  9494 

+0.844  2332 

+0.724  9743 

+0.848  9201 

+0.7348666 

48 

+0.7682082 

+0.690  9488 

+0.770  0409 

+0.705  9042 

+0.775  4674 

+0.7155486 

49 

+0.6974541 

+0.664  4077 

+0.7000152 

+0.679  2337 

+0.7044196 

+0.688  5925 

SO 

+0.629  8261 

+0.630  6012 

+0.632  2439 

+0.645  2428 

+0.636  5703 

+0.654  2871 

Si 

+0.5660715 

+0.5898843 

+0.568  3830 

+0.604  2938 

+0.572  6800 

+0.6130068 

52 

+0.5068987 

+0.542  6896 

+0.509  1472 

+0.5568286 

+0.5134653 

+0.565  2062 

53 

+0.452  9685 

+0.489  5225 

+0.455  2029 

+0.503  3636 

+0.459  5903 

+0.5114133 

54 

+0.404  8861 

+0.4309573 

+0.407  1576 

+0.444  4847 

+0.4116602 

+0.452  2270 

55 

+0.363  1948 

+0.3676314 

+0.365  5549 

+0.380  8404 

+0.3702130 

+0.388  3082 

56 

+0.328  3680 

+0.300  2365 

+0.330  8664 

+0.313  1351 

+0.335  7125 

+0.320  3708 

57 

+0.300  8042 

+0.2295127 

+0.303  4856 

+O.242  I2IO 

+0.308  5436 

+0.249  I75i 

58 

+0.280  8207 

+0.1562403 

+0.283  7236 

+O.I68  5895 

+0.289  0073 

+0.1755176 

59 

+0.268  6504 

+0.08  1  2302 

+0.271  8049 

+0.093  3621 

+0.277  3  1  70           +o.  loo  2240 

ACTION   OF  THE  PLANETS  ON   THE   MOON. 


TABLE   V .—Continued. 

RECTANGULAR  G-COORDINATES  X  AND  Y  OF  VENUS,  REFERRED,  IN  EACH 
SYSTEM,  TO  AN  Axis  OF  X  PASSING  THROUGH  THE  MEAN  SUN. 


I 

System  3. 

System  4. 

System  5. 

X 

r 

X 

T 

X 

J' 

o 

+0.279  8926 

+0.024  4020 

+0.285  2603 

+0.0184830 

+0.2886591 

+0.0080878 

I 

+0.284  2769 

—0.051  3907 

+0.289  6702 

—0.0569777        +0.2930191 

—0.067  0880 

2 

+0.296  5601 

—0.1262430 

+0.301  9298 

—0.131  4966        +0.305  1838 

—0.141  3542 

3 

+0.316  5918 

—0.1993308 

+0.321  8895 

—0.204  2627 

+0.325  0091 

—0.213  9093 

4 

+0.344  1354 

—0.269  8527 

+0.3493160 

—0.274  4873 

+0.352  2710 

—0.283  9728 

5 

+0.378  8721 

—0.337  0406 

+0.383  8974 

—0.341  4123 

+0.3866655 

—0.350  7937 

6 

+0.420  4078 

—0.400  1668 

+0425  2457 

—0404  3190 

+0.427  8152        —0413  6573 

7 

+0.468  2736 

—0458  5510 

+0472  9020 

—0.462  5340 

+0.475  2721        —0.471  8919 

8 

+0.521  9346 

—0.5115674 

+0.5263418 

—0.5154368 

+0.528  5222         —0.524  8756 

9 

+0.5807952 

-0.5586507 

+0.5849803 

—0.5624651 

+0.5869911        -0.5720427 

10 

+0.644  2076 

—0.5993022 

+0.648  1798 

—0.603  1207 

+0.650  0504 

—0.612  8887 

ii 

+0.7114758 

—0.633  0942 

+0.715  2549 

—0.636  9740 

+0.7170241 

—0.646  9765 

12 

+0.781  8659 

—0.659  6737 

+0.785  4819 

—0.663  6672 

+0.787  1933 

—0.673  9389 

13 

+0.854  6132 

-0.678  7647 

+0.858  1036 

—0.682  9185 

+0.859  8058 

—0.693  4842 

14 

+0.928  9294 

—0.690  1730 

+0.932  3397 

-0.694  5257 

+0.934  0825 

-0.705  3977 

IS 

+1.0040113 

—0.693  7852 

+1.0073925 

-0.698  3667 

+1.0092254 

—0.709  5459 

16 

+1.0790492 

—0.6895714 

+  1.0824565 

—0.694  4006 

+  1.0844261 

—0.705  8759 

17 

+1-1532351 

—0.677  5823 

+  1.1567246 

—0.682  6691 

+1.1588744 

—  0.6944182 

18 

+1.2257719 

-0.657  9526 

+  1.2293983 

—0.663  2956        +  1.23  1  7646 

—0.675  2864 

19 

+1.2958808 

—0.630  8962 

+  1.2996944 

—0.636  4839        +1.302  3064 

—0.648  6759 

20 

+1.3628093 

—0.596  7058 

+  1.3668546 

—0.6025160        +1-3697311 

—0.6148635 

21 

+14258394 

—0.555  7489 

+  1.4301522 

—0.561  7502 

+14333011 

—0.574  2026 

22 

+1.4842931 

—0.508  4047 

+14889006 

—0.5146175 

+1.4923178 

—O.527  1222 

23 

+1.5375412 

—0455  3593 

+1.5424596 

—0.461  6180 

+1.5461296 

—0474  1214 

24 

+1.5850084 

—0.397  0007 

+  1.5002425 

—0403  3157 

+  1.594  1387 

—0.415  7661 

25 

+1.6261801 

—0.334  0124 

+  1.631  7225 

—0.340  3342 

+  1.6358083 

—0.352  6827 

26 

+1.6606072 

—0.267  0689 

+1.6664388 

—0.273  3482 

+  1.6706690 

-0.285  5523 

27 

+1.6879113 

—0.1968867 

+  1.6940012 

—0.203  0779 

+  1.6983230 

—0.215  1035 

28 

+1.707  7882 

—0.1242202 

+  1.7140957 

—0.1302819 

+  1.7184514 

—0.142  1049 

29 

+  1.7200122 

—0.049  8508 

+  1.7264883 

—0.055  7493 

+  1.7308162 

—0.0673551 

30 

+1.7244388 

+0.0254188 

+  1.7310274 

+0.0197085           +1.7352660 

+0.008  3225 

31 

+1.7210059 

+0.100  7745 

+  1.7276465 

+0.095  2669 

+  I-73I  7363 

+0.084  0928 

32 

+1.7097359 

+0.175  3989 

+1.7163660 

+0.1700960 

+  1.7202518 

+0.1591154 

33 

+1.6907351 

+0.2484792 

+  1.6972920 

+0.243  3714           +1.700  9265 

+0.232  5558 

34 

+1.6641950 

+0.3192167 

+  1.6706195 

+0.314  2840 

+  1.6739638 

+0.303  5951 

35 

+1.6303893 

+0.3868336 

+1.6366271 

+0.382  0454 

+  1.6396529 

+0.371  4384 

36 

+1.5896721 

+0.450  5830 

+  1.5956761 

+0.445  9010 

+  1.5983674 

+0.435  3259 

37 

+1.5424753 

+0.509  7560 

+  1.5482079 

+0.505  1363 

+  1.5505612 

+0.4945411 

38 

+1.4893032 

+0.5636915 

+  14947388 

+0.5590849 

+  1.4967627 

+0.548  4197 

39 

+14307300 

+0.6117835 

+  14358543 

+0.607  1386 

+  1-437  5701 

+0.596  3565 

40 

+1.3673923 

+0.653  4880 

+  1.3722027 

+0.648  7530 

+1.373  6443 

+0.6378131 

41 

+1.2999821 

+0.688  3306 

+  1.3044890 

+0.683  4552 

+  1.3056998 

+0.672  3257 

42 

+1.2292412 

+0.7159113 

+  1.2334657 

+0.710  8508 

+  1.2344980 

+0.699  5075 

43 

+1.1559507 

+0.7359II4 

+  I.I599243 

+0.730  6274 

+  1.1608358 

+0.7190571 

44 

+1.0809236 

+0.748  0961 

+  1.0846868 

+0.742  5595 

+  1.0855389 

+0-730  7598 

45 

+1.0049938 

+0.752  3188 

+  1.0085958 

+0.7465108 

+  1.0094492 

+0.7344915 

46 

+0.929  0085 

+0.748  5233 

+0.932  5029 

+0.742  4365 

+0.933  4172 

+0.7302183 

47 

+0.853  8157 

+0.736  7446 

+0.857  2595 

+0.730  3837 

+0.8582903           +0.7179976 

48 

+0.780  2560 

+0.717  1082 

+0.783  7064 

+0.7104897 

+0.784  9033           +0.697  9763 

49 

+0.709  1529 

+0.6898295 

+0.712  6639 

+0.682  9817 

+0.7140686           +0.6703895 

50 

+0.641  3017 

+0.6552115 

+0.644  9229 

+0.648  1733 

+0.646  5668           +0.635  5557 

5i 

+0.5774611 

+0.6136402 

+0.581  2344 

+0.606  4605 

+0.583  1378           +0.593  8734 

52 

+0.5183431 

+0.565  5804 

+0.522  3027 

+0.5583153 

+0.5244736 

+0.545  8162 

53 

+0464  6055 

+0.5115714 

+0.468  7755 

+0.504  2817 

+0.471  2105 

+0.491  9264 

54 

+0.4168456 

+04522194 

+0.421  2379 

+0.444  9681 

+0.423  9224 

+0.432  8084 

55 

+0.375  5907 

+0.388  1916 

+0.380  2062 

+0.381  0400 

+0.3831147 

+0.369  1218 

56 

+0.341  2944 

+0.320  2069 

+0.346  1218 

+0.3132132 

+0.349  2206           +0.301  5727 

57 

+0.3143306 

+0.249  0281 

+0.3T93477 

+0.242  2440 

+0.3225952           +0.2309071 

58 

+0.204  9887 

+0.175  4508 

+0.300  1638 

+0.1689220 

+0.303  5124           +0.157  9020 

59 

+0.283  4712 

+0.1002967 

+0.288  7653 

+0.094  0597 

+0.2921646           +0.0833577 

COEFFICIENTS  FOR  DIRECT  ACTION. 


57 


TABLE   V '.—Continued. 

RECTANGULAR  G-COORDINATES  X  AND  Y  OF  VENUS,  REFERRED,  IN  EACH 
SYSTEM,  TO  AN  Axis  OF  X  PASSING  THROUGH  THE  MEAN  SUN. 


i 

System  6. 

System  7. 

System  8. 

X 

r 

X 

r 

X 

r 

o 

+0.2893812 

—0.004  3547 

+0.287  0415 

—0.0159849 

+0.281  8724 

—0.023  8026 

i 

+0.293  5931 

-0.0793938 

+0.291  OOI2 

—0.091  0510 

+0.285  5340 

—0.099  0059 

2 

+0.305  5780 

-0.153  578i 

+0.302  7254 

—0.1653184 

+0.2969815 

—0.1734542 

3              +0.32S  2014 

—  0.226  1096 

+0.322  0914 

—0.2379885 

+0.316  1026 

—0.246  3419 

4              +0.352  2490 

—0.296  2092 

+0.348  8955 

—0.308  2783 

+0.342  7024 

-0.3168787 

5              +0.3864284 

—0.363  1251 

+0.382  8548 

-0.375  4297 

+0.376  5054 

—0.384  2966 

6             +0.427  3728 

—0.426  1400 

+04236113 

—0438  7161 

+0.417  1580 

-0.4.178587 

7             +0.474  6443 

—0.484  5776 

+0470  7341 

—04974515 

+0.464  2316 

—0.5068671 

8             +0.527  738o 

-0.537  8098 

+0.523  7240 

—0.5509962 

+0.5172268 

—0.560  6696 

9             +0.586  0872 

—0.585  2627 

+O.5820I78 

—0.598  7649 

+0.575  5777 

-0.6086687 

10             +0.649  0693 

—0.6264213 

+0.6440946 

—0.640  2315 

+0.6386584 

-0.650  3275 

n              +0.7160118 

-0.6608369 

+O.7II  9802 

—0.674  9360 

+0.705  7897 

—0.685  1772 

12             +0.786  1986 

—0.688  1308 

+0.782  2554 

—0.702  4883 

+0.776  2436 

—0.7128211 

13             +0.8588769 

—0.707  9983 

+0.855  0617 

—0.722  5740 

+0.849  2528 

—0.732  9398 

14 

+0.9332651 

—0.720  2135 

+0.929  6105 

—0.734  9582 

+0.924  0172 

—0.745  2958 

15 

+1.0085598 

—0.724  6313 

+1.0050902 

—0.7394890 

+0.099  7145 

-0.7497373 

16 

+1.0839453 

—0.721  1895 

+  1.0806753 

—0.736  0095 

+1.075  5o85 

—0.746  1991 

17 

+1.1586020 

—0.709  9106 

+  i.  155  5347 

—0.724  8102 

+I.I505579 

—0.734  7062 

18 

+1.231  7142 

—0.6909018 

+  1.2288424 

—0.705  7280 

+1.2240268 

-0.715  3728 

19 

+1.3024807 

—0.664  3547 

+  1.2997857 

—0.679  0484 

+1.2950946 

—  0.688  4024 

20 

+1.3701219 

—0.630  5443 

+  1-3675740 

—0.645  0509 

+1.3629645 

—0.654  0862 

21 

+1.4338001 

—0.589  8264 

+  1.4314510 

—0.604  0981 

+14268752 

—0.612  7087 

22 

+1.4930762 

—0.542  6339 

+  1.4907009 

—0.5566323 

+  1.4861086 

—0.564  9945 

23 

+1.5470187 

—0.4894723 

+  1.5446576 

—0.503  1709 

+1.5399987 

—0.5112033 

24 

+1.5951122 

—0.4309158 

+  1.5927131 

—0.444  2908 

+1.5879409 

—0.452  0237 

25 

+1.6368140 

—0.367  6003 

+  1.6343252 

—0.3806684 

+1.6293973 

—0.388  1  167 

26 

+1.6716503 

—0.3002166 

+  1.6690214 

—0.312  9805 

+1.6639048 

—0.320  1963 

27 

+  1.6992228 

—0.229  5034 

+  1.6964089 

—0.241  9865 

+1.691  0789 

—0.249  0223 

28 

+1.7192138 

—  0.1562412 

+  1.716  1761 

—0.1684765 

+  1.7106192 

-0.1753907 

29 

+1.731  3808 

—0.081  2404 

+  1.7280986 

—0.093  2689 

+  1.7223122 

—  o.ioo  1249 

30 

+1.7356043 

—0.005  3345 

+  1.7320410 

—0.0172049 

+  1.7260340 

—  0.024  0644 

31 

+1.7318010 

+0.070  6309           +1.727  958o 

+0.0588662 

+1.721  7514 

+0.0519430 

32 

+1.7200145 

+0.1458077           +1.7158966 

+0.1340934 

+1.7095212 

+0.1270511 

33 

+  1.7003702 

+0.2  19  3560           +i  .695  0944 

+0.207  6356 

+1.6894904 

+0.2004263 

34 

+1.673  0827 

+0.290  4526 

+  1.6684777 

+0.278  6726 

+1.6618933 

+0.271  2560 

35 

+1.6384553 

+0.358  3016 

+  1.6336507 

+0.3464130 

+1.6270482 

+0.3387587 

36 

+1.5968743 

+0.422  1439           +I-59I  9353 

+0.410  1049 

+1.5853540 

+0.402  1930 

37 

+1.5488048 

+0.4812659           +I.S437778 

+0469  0434 

+1.5372837 

4-0.4608658 

38 

+1.4947865 

+0.535  0089           +  1  489  7309 

+0.522  5796 

+14833802 

+0.5141397 

39 

+1.4354258 

+0.582  7755           +i  .430  4045 

+0.570  1266 

+  1.4242480 

+0.561  4397 

40 

+1.3713900 

+0.6240380          +1.3664651 

+0.611  1675 

+1.3605472 

+0.602  2588 

41 

+1.303  3980 

+0.6583431 

+  1.2986290 

+0.645  2002 

+1.2929865 

+0.636  1634 

42 

+1.2322125 

+0.685  3171 

+  1.2276529 

+0.672  0417 

+1.2223115 

+0.662  7978 

43 

+1.1586310 

+0.7046608           +1.1543266 

+0.691  2326 

+  1.1493005 

+0.681  8865 

44 

+1.0834752 

+0.716  1979 

+  1.0794622 

+0.702  6377 

+  1.0747540 

+0.693  2381 

45 

+1.0075811 

+0.7197864 

+  1.0038858 

+0.706  1489 

+0.9094847 

+0.696  7453 

46 

+0.931  7009 

+0.7154100 

+0.9284277 

+0.701  7444 

+0.9243115 

+0.692  3855 

47 

+0.8569412 

+0.703  1324 

+0.853  9128 

+0.689  4895 

+0.850  0486 

+0.680  2209 

48 

+0.7838553 

+0.683  1047 

+0.781  1516 

+0.669  5340 

+0.7774975 

+0.660  3960 

49 

+0.713  3338 

+0.655  5634 

+0.7109317 

+0.642  1  1  1  1 

+0.7074383 

+0.633  1359 

50 

+0.646  1446 

+0.620  8269 

+0.644  0104 

+0.607  5322 

+0.640  6223 

+0.558  7436 

Si 

+0.5830165 

+0-579  2912 

+0.581  1065 

+0.566  1862 

+0.577  7638 

+0.5575971 

52 

+0.524  6297 

+0.5314234 

+0.522  8929 

+0.5185316 

+0.5195340 

+0.510  1441 

53 

+0.471  6  no 

+0.4777588 

+04699907 

+0.465  0927 

+0466  5545 

+04568979 

54 

+0.424  5246 

+0.418  8929 

+0.422  9616 

+0.4064538 

+0.4193902 

+0.3084327 

55 

+0.383  8706 

+0.355  4747 

+0.382  3050! 

+0.343  2525 

+0.378  5456 

+0.335  3763 

56 

+0.350  0760 

+O.288  2O02 

+0.3484501! 

+0.276  1738 

+0.3444581 

+0.2684051 

57 

+0.323  4952 

+0.2178039 

+0.321  7545 

+0.205  9419 

+0.3174943 

+0.1982368 

58 

+0.3044032 

+O.I450SI2 

+0.302  4906 

+0.133  3140 

+0.2079461 

+0.1256232 

59 

+0.292  9944 

+0.070  7305 

+0.2908879, 

+0.0590724 

+0.286  0278 

+0.051  3438 

ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE  V .—Concluded. 

RECTANGULAR  G-COORDINATES  X  AND  Y  OF  VENUS,  REFERRED,  IN  EACH 
SYSTEM,  TO  AN  Axis  OF  X  PASSING  THROUGH  THE  MEAN  SUN. 


i 

System  9. 

System  10. 

System  n. 

X 

r 

X 

r 

X 

r 

o 

i 

2 

3 

4 

+0.275  0523 
+0.2784844 
+0.2897521 
+0.308  7484 
+0.335  2831 

-0.025  3633 

—  o.ioo  7462 
—0.1754000 
—0.248  5086 
—0.3192704 

+0.268  5973 
+0.271  9768 
+0.283  2537 
+0.302  3220 
+0.328  0878 

—0.019  7819 
—0.095  3562 
—0.1702047 
—0.243  5018 
—0.3144354 

+0.264  6365 
+0.268  1640 
+0.279  6432 

+0.298  9610 

+0.325  9149 

—0.0084260 
—0.084  2045 
—0.1592361 
—0.232  6860 

-0.303  7353 

5 
6 

8 
9 

+0.369  0820 
+0.409  7903 
+0.4569758 
+0.510  1331 

+0.568  6886 

—0.386  9062 
—0.450  6682 
—0.509  8485 
-0.563  7854 

—0.6118737 

+0.362  9727 
+0403  9145 
+0.451  3723 
+0.504  8294 
+0.563  7000 

—0.3822158 
—04460864 
—0.5053317 
—0.559  2848 
—0.607  3382 

+0.3602162 
+0401  4913 

+0.4492861 
+0.503  0724 
+0.5622517 

—0.371  5891 
—0.435  4874 
—0.494  7126 
-0.548  5982 
—0.596  5386 

10 
ii 

12 

13 
14 

+0.632  0062 
+0.6993951 
+0.7701160 
+0.8433898 
+0.918  4056 

-0.653  5698 

•   —0.688  4000 
—0.7159656 
—0.735  9488 
—0.748  1159 

+0.6273358 
+0.695  0329 
+0.766  0396 
+0.839  5657 
+0.914  7912 

—0.648  0481 
—0.683  6429 
—0.711  0281 
—0.730  7919 
—0.742  7091 

+0.626  1636 
+0.694  0946 
+0.765  2841 
+0.838  9356 
+0.914  2243 

—0.637  9955 
—0.672  5042 
—0.6996795 
—0.719  2203 
—0.7309118 

3 

17 

18 

19 

+0-994  3301 
+  1.0703171 
+  I.I455I78 
+1.2190910 
+1.2902123 

—0.752  3209 
—0.748  5084 
—0.736  7126 
—0.717  0597 
-0.689  7649 

+0.990  8753 
+  1.0669676 
+  1.1422168 
+  1.2157813 
+  1.2868399 

—0.7466441 
—0.742  5523 
—0.730  4812 
—0.7105689 
—0.683  0438 

+0.990  3091 
+1.0663406 
+1.1414717 
+1.2148681 
+1.2857175 

-0.734  6303 
—0.730  3419 
—  0.718  1042 
—0.698  0645 
—0.670  4585 

20 
21 
22 
23 

24 

+  1.3580846 
+  1.4210480 
+  1.481  0876 
+  1.5348440 
+  1.5826192 

—0.655  1320 
—0.613  5482 
—0.565  4803 
—0.511  4688 

—  O.452  I2IO 

+  1-3545999 
+  1.4183087 
+  1.4772619 
+  1.5308092 
+  1.5783646 

—0.6482193 
—0.6064921 

—0.558  3354 
—0.504  2948 
—0.444  9805 

+1.3532374 
+14166861 
+1.4753700 
+1.5286519 
+  1-5759570 

—0.635  6060 
—0.593  9072 
—0.545  8365 
—0.491  9379 
—0432  8169 

25 
26 
27 

28 
29 

+  1.6238856 
+  1.658  1899 
+  1.685  1598 
+  1.7045055 
+  1.7160249 

—0.388  1040 
—0.320  1363 
—0.2489795 
—0.175  4284 
—0.1003034 

+  1.6194111 
+  1.6535064 
+  1.6802882 
+  1.6994769 
+  1.7108776 

—0.381  0500 
—0.3132470 
—0.242  3001 
—0.1690056 
—0.094  1744 

+  1.6167795 
+  1.6506865 
+  1.6773221 
+  1.6964120 
+  1.7077640 

—0.369  1332 
-0.301  5947 
—0.230  9478 
—0.1570687 
—0.083  456i 

30 
3i 
32 
33 

34 

+  1.7196036 
+  1.7152172 
+  1.7029292 
+  1.6828904 
+  1.6553370 

—0.024  4386 
+0.051  3259 
+0.126  1551 
+O.I992270 
+0.269  7422 

+  1.7143826 
+  1.7099702 
+  1.6977055 
+  1.6777386 
+  1.6503021 

—0.0186200 
+0.056  8022 
+0.131  2957 
+0.204  0425 
+0.274  2547 

+  1.7112698 
+  1.7060075 
+  1.6947377 
+  1.6740043 
+  1.6476325 

—0.008  2204 
+0.066  9225 
+0.141  1607 
+0.213  6945 
+0.283  7444 

35 
36 
37 
38 
39 

+  1.6205873 
+  1.5790366 
+  I-53I  1536 
+  14774737 
+  1.4185926 

+0.3369322 
+0.400  0673 
+0.4584650 
+0.5II4975 

+0.558  5979 

+  1.6157077 
+  1-5743442 
+  1.5266706 
+  14732125 
+  I.4I45550 

+0.341  1756 
+0.404  0865 
+04623116 
+0.5152296 
+0.562  2757 

+  1.6132252 
+  1.5720603 
+  1.5245861 
+  14713167 
+  14128279 

+0.3505588 
+0.413  4229 
+0.471  6640 
+0.524  6595 
+0.571  8423 

40 
41 
42 
43 

44 

+  1.355  1605 
+  1.2878735 
+  1.2174671 
+  1.1447074 
+  1.0703837 

+0.599  2658 
+0.633  0725 
+0.659  6638 
+0.678  7636 
+0.690  1770 

+  I.35I337I 
+  1.2842449 
+  1.2140031 
+  1.141  3690 
+  1.067  1246 

+0.602  9494 
+0.636  8197 
+0.663  5280 
+0.682  7924 
+0.694  4103 

+  1.3497496 
+  1.2827595 
+  1.2125774 
+  I.I399554 
+  1.0656731 

+0.612  7065 
+0.646  8129 
+0.673  7929 
+0.693  3538 
+0.705  2806 

4| 
46 

47 
48 
49 

+0.995  2985 
+0.920  2620 
+0.8460814 
+0.773  5541 
+0.703  4583 

+0.693  7915 
+0.689  5775 
+0.6775884 
+0.657  9598 
+0.630  9076 

+0.992  0676 
+0.9170042 
+0.842  7410 
+0.7700760 
+0.699  7507 

+0.6982589 
+0.6042980 
+0.682  5604 
+0.663  1965 
+0.636  3845 

+0.090  5284 
+0.915  3290 
+0.8408858 
+0.768  0039 
+0.6974731 

+0.709  4400 
+0.705  7790 
+0.604  3282 
+0.675  2018 
+0.648  5938 

50 
51 

52 

53 

54 

+0.636  5464 
+0.573  5356 
+0.515  1027 
+0.461  8752 
+0.4144269 

+0.506  7255 
+0-555  7804 
+0.5085110 
+0.455  4212 
+0.397  0772 

+0.632  6437 
+0.5693612 
+0.5106299 
+o.  157  0893 
+0409  3243 

+0.6024170 
+0.561  6533 
+0.5145258 
+0461  5341 
+0.403  2415 

+0.6300613 
+0.566  5049 
+0.507  5018 
+0.453  7038 
+0405  7084 

+0.614  7816 
+0.574II95 
+0.527  0362 
+0474  0329 
+04156762 

55 
56 
57 
58 
59 

+0-373  2719 
+0.3388588 
+0.311  5656 
+0.291  6064 
+0.2794770 

+0.334  10  12          +0.367  86  1  1 
+0.267  1652          +0.333  1597 
+0.196  9846          +0.305  6091 
+0.124  3117          +0.285  5228 
+0.049  9280          +0.273  1348 

+0.340  2695 
+0.273  2007 
+0.203  0240 
+0.1302265 
+0.055  6874 

+0.364  0524 
+0.329  2053 
+0.301  5630 

+O.28l  W/\7. 
+0.209  0848 

+0.352  5927 
+0.285  4627 
+0.215  0136 
+0.1420126 
+0.067  2585 

COEFFICIENTS  FOR  DIRECT  ACTION. 


59 


TABLE  VI. 

G-COORDINATE    Z    OF    VENUS. 


Sys- 
tem 
i 

o 

I 

2 

3 

4 

5 

0 

+.0174641 

+.034  6635 

+.0423808 

+.038  5648 

+.0243031 

+.003  5232 

i 

+  .021  4657 

+.037  0784 

+.042  5597 

+.0364563 

+.0205041 

—.0009384 

2 

+.025  2276 

+.0390785 

+.042  2557 

+.033  9453 

+.0164810 

—.005  3894 

3 

+.028  7075 

+.0406417 

+.041  4818 

+.0310603 

+.OI2  2785 

—.009  7824 

4 

+.03I866S 

+.041  7509 

+.040  2471 

+.027  8339 

+  .0079423 

—.014  0692 

5 

+  .0346689 

+.0433939 

+.0385664 

+.024  3022 

+.003  5202 

—.018  2044 

6 

+  .0370834 

+.042  5642 

+.036  4589 

+.020  5043 

—  .OOO  94OO 

—.022  1436 

7 

+  .0390828 

+.042  2603 

+.033  9486 

+.0164824 

-.005  3898 

—.025  8445 

8 

+.040  6450 

+.041  4861 

+.031  0641 

+.0122809 

—.009  7816 

—.029  2675 

9 

+  .041  7527 

+.040  2507 

+.0278378 

+.0079454 

—  .014  0672 

—.032  3763 

10 

+.042  3943 

+.038  5650 

+.024  3057 

+.003  5235 

—.018  2018 

—•035  1374 

ii 

+.042  5630 

+.0364602 

+.0205072 

—.000  9368 

—.022  1406 

—.037  5214 

12 

+.0422579 

+.033  9485 

+.0164842 

—.005  3871 

—  .025  8416 

—•039  5028 

13 

+.041  4828 

+.031  0627 

+.012  2814 

—.009  7798 

—.0292651 

—.041  0601 

U 

+.040  2473 

+.0278355 

+.007  9446 

—.014  0665 

—.032  3746 

—.042  1764 

15 

+.038  5660 

+.024  3029 

+.003  5218 

—.Ol8  2O22 

—.035  1366 

—.042  8392 

16 

+.036  4580 

+.020  5046 

—.000  9389 

—.022  1419 

—•037  5215 

—.043  0410 

17 

+.033  9476 

+.016  4822 

—.005  3893 

—.025  8434 

-.039  5037 

—.042  7792 

18 

+.0310631 

+.012  2804 

—.009  7815 

—.029  2671 

—.041  0620 

—.042  0556 

19 

+.027  8370 

+.007  9450 

—.0140675 

—.032  3762 

—.042  1783 

-.040  877S 

20 

+.0243054 

+.003  5233 

—.0182021 

—.035  1377 

—.042  841  1 

—.039  2568 

21 

+.020  5072 

—.000  9367 

—.022  1407 

—.037  5218 

—.043  0424 

—.037  2103 

22 

+.0164847 

—.005  3866 

—.025  8415 

—•039  5032 

—.042  7800 

—•034  7593 

23 

+  .OI2  2824 

—.009  7789 

—.029  2648 

—.041  0604 

—.042  0556 

—.031  9296 

24 

+  .0079461 

—.0140654 

—  •0323739 

—.042  1765 

—.040  8769 

—.0287515 

25 

+.003  5237 

—.018  2007 

—•035  1357 

—.042  8391 

-.039  2556 

—.025  2500 

26 

—.000  9370 

—.022  1401 

—.037  5204 

—.043  0406            —.037  2089 

—.021  4899 

27 

—.005  3875 

—.025  8415 

—.039  5023 

—.042  7786            —.034  7578 

—.0174848 

28 

—.009  7799 

—.029  2652 

—.041  0600 

—.042  0548 

—.031  9285 

—.OI3  2876 

29 

—.0140663 

—.032  3746 

—.042  1766 

—.040  8766 

—.028  7508 

—.0089440 

30 

—.0182014 

—.035  1364 

—.042  8395 

—.0392557 

—.025  2588 

—.0015014 

31 

—  .022  1404 

—.037  5209 

—.0430411 

—.037  2091 

—.021  4900 

—.0000091 

32 

—.025  8416 

—.039  5026 

—.042  7789 

-.0347581 

—.0174851 

+.004  4833 

33 

—  .0292651 

—.041  0601 

—.042  0550 

—.031  9287 

—.0132878 

+.0089260 

34 

—.032  3744 

—.042  1765 

—.040  8765 

—.028  7508 

—.008  9438 

+.013  2696 

35 

—•035  1363 

—.042  8394 

—•039  2555 

-.025  2585 

—.004  5009 

+.0174653 

36 

—.0375212 

—.0430411 

—.037  2089 

—.021  4895 

—  .0000084 

+.0214663 

37 

—•039  5032 

—.042  7792 

—.0347581 

—  .0174846 

+.004  4842 

+.025  2276 

38 

—.0410611 

—.042  0556 

—.0319288 

—.013  2874 

+.0089269 

+.028  7069 

39 

—.042  1777 

-.0408775 

-.0287513 

—.0080438 

+.013  2702 

+.0318651 

40 

—.042  8407 

—.0392567 

—.025  2504 

—.004  5015 

+.0174654 

+.0346668 

41 

—.043  0424 

—.037  2103 

—.0214908 

—.000  0095 

+.021  4658 

+.0370806 

42 

—.042  7801 

—•034  7594 

—.0174861 

+.004  4827 

+.O25  2265 

+.0390797 

43 

—.042  0559 

—.031  9300 

—  .013  2889 

+.0089251 

+.028  7052 

+.0406420 

44 

—.0408771 

—.0287519 

—.0089451 

+.013  2684 

+.031  8631 

+.041  7501 

45 

—•039  2555 

—.025  2594 

—.004  5022 

+.0174641 

+.034  6648 

+.042  3922 

46 

—.037  2085 

—.021  4900 

—  .0000095 

+.021  4650 

+.0370791 

+.042  5618 

47 

—.034  7572 

—.0174846 

+.004  4835 

+.025  2265 

+.0390788 

+.042  2575 

48 

—.031  9278 

—.0132869 

+.0089269 

+.028  7061 

+.0406418 

+.041  48.34 

49 

—.028  7504 

—.0084931 

+.013  2707 

+.031  8649 

+.041  7508 

+.0402487 

50 

—.025  2588 

—.004  5007 

+.017  4664 

+.034  6672 

+.042  3939 

+.038  5680 

Si 

—.021  4907 

—.0000088 

+.02  1  4671 

+.037  0818 

+.042  5641 

+.036  4606 

52 

-.0174867 

+.004  4829 

+.025  2277 

+.0390815 

+.042  2603 

+•033  9503 

53 

—.013  2004 

+.008  9247 

+.028  7061 

+.040  6439 

+.0414862 

+.031  0659 

54 

—.008  9473 

+.013  2672 

+.031  8635 

+.041  75i8 

+.0402511 

+.0278396 

55 

—.004  5052 

+.0174621 

+.0346645 

+.042  3935 

+.038  5697 

+.024  3076 

56 

—.0000130 

+  .021  4624 

+.0370779 

+.042  5624 

+.036  461  1 

+.020  5092 

57 

+.004  4797 

+.025  2232 

+.0390768 

+.042  2573 

+-°33  9495 

+.0164863 

58 

+.008  9228 

+.028  7024 

+.0406300 

+.041  4822 

+.0310636 

+.012  2834 

59 

+.013  2669 

+.031  8610 

+.041  7474 

+.0402465 

+.0278361 

+.007  9465 

6o 


ACTION   OF  THE   PLANETS  ON   THE  MOON. 


TABLE   VI.— Concluded. 

G-COORDINATE    Z    OF    VENUS. 


Sys- 
tem 
i 

6 

7 

8 

9 

10 

II 

0 

—  .0182000 

—.035  1342 

—.042  8369            —.039  2532 

—.025  2572 

—.004  5010 

i 

—.022  1401 

—.037  5194 

—.043  0386            —  .037  2062 

—  .021  4874 

—.000  0076 

2 

—.025  8423 

—.039  5018 

-.042  7767             —-034  7550 

—.0174818 

+.0044858 

3 

—.029  2668 

—.041  0601 

—.042  0532             —  .031  9256 

—.013  2840 

+.0089294 

4 

—.032  3768 

—.042  1773 

—  .040  8754             —  .028  748  1 

—.008  9400 

+.013  2734 

5 

—•035  1389                 —.042  8410 

—•039  2552             —.025  2564 

—.004  4976 

+.0174692 

6 

—.037  5236            —.043  0430 

—  .037  2094             —  .021  4882 

—  .000  0058 

+.021  4697 

7 

—.039  5053 

—.042  7812 

—•0347591              —.0174841 

+.004  4860 

+.025  2302 

8 

—.041  0626 

—.0420571 

—.031  9302             —.013  2877 

+.0089278 

+.028  7085 

9 

—.042  1786 

—.040  8785 

—.028  7525             —.008  9446            +.013  2704 

+.031  8658 

10 

—.0428409 

—.039  2571 

—.0252602             —.0045023            +.0174650 

+.0346667 

ii 

—.0430419 

—.0372100 

—.0214909            —.000  oioo           +.0214652 

+  .0370801 

12 

—.042  7792 

—.034  7584 

—.017  4855           +.004  4828 

+.025  2260 

+  .0390788 

13 

—.042  0547 

—.031  9285 

—.013  2877            +.008  9260 

+.028  7051 

+.O4O  64IO 

14 

-.0408759 

—.028  7504 

—.0089434 

+.0132608 

+.031  8637 

+.041  7493 

15 

—.0392548 

—.025  2577 

—.004  5004 

+.0174658 

+.034  6659 

+.042  3920 

16 

—.0372083 

—.021  4886 

—.0000078            +.0214667 

+.037  0804 

+.042  5621 

17 

-•034  7576 

—.0174838 

+  .004  4849           +  .025  2279 

+.039  0801 

+.042  2582 

18 

—.031  9285 

—.013  2868 

+.008  9276           +.028  7070 

+.040  6429 

+.041  4843 

19 

—.0287511 

-.0089433 

+.013  2707           +.031  8652 

+.041  7515 

+.0402494 

20 

—.025  2591 

—.004  5010 

+.0174659          +.0346669 

+.0423938    !    +.0385683 

21 

—.021  4905 

—.0000089 

+.021  4663                 +0.37  0806 

+.042  5634 

+.0364601 

22 

—.0174858 

+.004  4834 

+.025  2269             +.039  0798 

+.042  2588 

+.033  9492 

23 

—.013  2884 

+.0089260 

+.028  7058            +.040  6420 

+.041  4841 

+.031  0639 

24 

—.0089445 

+.013  2693 

+.031  8639 

+.041  7503 

+.040  2485 

+.0278371 

25 

—.004  5016 

+.017  4648 

+.034  6657           +.042  3926 

+.038  5671 

+.024  3046 

26 

—  .0000090 

+.021  4657 

+.037  0798           +.042  5622 

+.036  4589 

+.020  5061 

27 

+.004  4836 

+.025  226> 

+.0390792          +.0422578 

+.033  9480 

+.0164835 

28 

+.0089263 

+.028  7059 

+.0406418           +.0414834 

+.031  0630 

+.OI2  28l2 

29 

+.013  2697 

+.031  8640 

+.0417503          +.0402483 

+.0278365 

+.0079453 

30 

+.0174651 

+  .0346658                 +.0423926          ,         +.0385671 

+.024  3044 

+.003  5232 

31 

+.02  1  4656 

+  .0370798 

+.042  5622           +.036  4589 

+.020  5061 

—.000  9372 

32 

+.025  2265 

+.0390700 

+.042  2577             +.033  948i 

+.0164835 

—.005  3874 

33 

+.028  7055 

+.O4O  6414 

+.041  4832             +.031  0629 

+.OI2  2813 

-.009  7798 

34 

+.031  8637 

+.041  7499 

+.040  2479             +.027  8362 

+.0079452 

—  .014  0663 

35 

+.034  6657 

+.042  3924 

+.038  5666 

+.0243039 

+.003  5227 

—.0182016 

36 

+.037  0803 

+.042  5624 

+.0364587 

+.020  5053 

—  .000  9380 

—  .022  1410 

37 

+.039  0800 

+.0422582 

+.033  948i 

+.OT6  4829 

-.005  3885 

—.025  8425 

38 

+.040  6429 

+.041  4843 

+.031  0634 

+.012  28lO 

—.0097809 

—.029  2663 

39 

+.041  75i6 

+.040  2496 

+.027  8373 

+.007  9454 

—.014  0673 

—.032  3757 

40 

+.042  3941 

+.038  5686 

+.024  3059 

+.003  5236 

—  .0182018 

-.035  1375 

41 

+.042  5637 

+.0364607 

+.020  5076 

—.000  9364 

—.022  1405 

—.037  5218 

42 

+.042  2590 

+.033  9500 

+.0164854 

—.005  3861 

—.0258411 

—.039  5031 

43 

+.041  4843 

+.031  0647 

+.012  2833 

—.009  7782 

—.029  2643 

—  .041  0603 

44 

+.040  2488 

+.0278377 

+.007  9472 

—.014  0645 

—.032  3732 

—  .042  1761 

45 

+.0385672 

+.0243051 

+.003  5245 

—.018  1997 

—.035  1349 

—.042  8386 

46 

+.0364590 

+.020  5062 

—  .000  9367 

—.022  1394 

—•0375197 

—.043  0401 

47 

+.033  948o 

+.0164835 

-.005  3875 

—.025  8414 

—.039  5020 

—.042  7782 

48 

+.031  0631 

+.0122812 

—.009  7805 

—.029  2657 

—.041  0604 

—  .042  0548 

49 

+.0278367 

+.007  9452 

—.0140671 

—.032  3758 

—.042  1777 

—.040  8770 

50 

+.0243049 

+.0035231 

—.018  2023 

-.035  1381 

—.042  8413 

—.0392569 

51 

+JO20  5O7I 

—.000  9371 

—.022  1413 

—.037  5229 

—.043  0437 

—.0372112 

52 

+.0l6  4852 

—.005  3869 

—.025  8422 

—.039  5045 

—0.42  7820 

—.034  7610 

53 

+.012  2837 

—.009  7786 

—.029  2654 

—.041  0619 

—.042  0581 

—.031  9323 

54 

+.007  9482 

—  .0140642 

—.032  3741 

—.042  1778 

—.0408794 

—.028  7548 

55 

+.003  5264 

—.018  1987 

—.035  1353 

—.042  8401 

—.039  2580 

—  .025  2624 

56 

—.000  9340 

—  .022  1376 

—.0375193 

—.0430411 

—.037  2108 

—.021  4931 

57 

—.005  3844 

-.025  8388 

—.039  5006 

—.042  7784 

—.034  7591 

-.0174875 

58 

—.009  7771 

—.0292624 

—.041  0578 

—.042  0538 

—.031  9290 

—.OI3  2804 

59 

—.014  0642 

—.032  3720            —.042  1740 

—.040  8748 

—.028  7503 

—.0089447 

COEFFICIENTS   FOR  DIRECT  ACTION. 


6l 


TABLE   VII. 
VALUES  OF  A,  B,  C,  AND  D  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


System  o. 

Sjstem  i. 

1 

A 

B 

C 

D 

i 

A 

B 

C 

D 

0 

+35.541  47 

—17.87647 

-17.66499    +  1.07427 

o 

+32-593  73 

—16.607  73 

-15.98599 

+  3-18745 

i 

+27.684  74 

—12.48321 

—15.20155    —11.36932 

i 

+27.372  10 

-13.08833 

—14.28378 

—  9-17780 

2 

+  14.10943 

-  3-81823 

—10.291  18     —12.81204 

2 

+  14.89321 

-  4-857  J7 

—10.03602 

—11.99256 

3 

+  6.07222 

+  0.26235 

—  6.33456     —  9.16752 

3 

+  6.704  15 

—  0.37485 

—  6.32930 

-  9.06478 

4 

+  2.63936 

+  1-25133 

-  3-89071     -  5.85772 

4 

+  3-00334 

+  0.935  15 

—  3.93850 

-  5-93021 

5      +  1.27090 

+  1.20398 

—  2.47487     —  3-74692 

5 

+  147429 

+  1.04697 

—  2.52125 

—  3.830  19 

6 

+  0.70797 

+  0.93776 

—  1.64574     —  2.48604 

6 

+  0.82746 

+  0.85385 

—  1.68130 

—  2.54954 

7 

+  0.45996 

+  0.68285 

—    I.I428I    !    —    1.72274 

7 

+  0.53500 

+  0.63364 

—  1.16865 

—  1.76704 

8     +  0.341  13 

+  0.48405 

—   0.82517       —    I.2433I 

8 

+  0.391  31 

+  045253 

-0.84383 

—  1.27360 

p 

+  0.27866 

+  0.33795 

—  0.61661 

—  0.92936 

9 

+  0.31404 

+  0.316  17 

—  0.63022 

—  0.95006 

10 

+  0.24248 

+  0.23228 

-  0.47476  \  —  0.71540 

10 

+  0.26852 

+  0.21633 

—  0.484  84 

—  0.72959 

ii     +  0.21941 

+  0.15584 

—  0.37525     —  0.56421 

ii 

+  0.23923 

+  0.14361 

—  0.382  83 

—  0.57395 

12 

+  0.20337 

+  o.ioo  16 

—  0.30352 

—  0.45390 

12 

+  0.21885 

+  0.09046 

—  0.30932 

—  046055 

13 

+  0.191  38 

+  0.05922 

—   0.250  58       —   0.371  12 

13 

+  0.20373 

+  0.051  35 

—  0.255  07 

—  0.375  58 

14 

+  0.181  go 

+  0.02881 

—  0.21071     —  0.30744 

14 

+  0.191  91 

+  0.02231 

—  0.21422 

—  0.31032 

15 

+   0.174  12 

+  0.00601 

—  0.180  13    —  0.25736 

15 

+  0.18233 

+  0.00058 

—  0.18250 

—  0.259  10 

16 

+   0.16756 

—  o.on  23 

—  0.15632    —  0.217  19 

16 

+  0.17435 

—  0.01582 

—  0.158  53 

—  0.21809 

17 

+  0.161  93 

—  0.02438 

—  0.13755    —  0.18439 

17 

+  0.16757 

—  0.02827 

—  0.13930 

—  0.18466 

18 

+  0.15705      —  0.03447 

—  0.122  s8     —  0.157  18 

18 

+  0.161  75 

—  0.03778 

—  0.123  98 

—  0.15697 

19 

+  0.15280 

—  0.04226     •-  0.11054     —  0.13423 

19 

+  0.15672 

—  0.04508 

—  O.HI  65 

—  0.13368 

20 

+  0.14909 

—  0.04831  ;  —  0.10078     —  0.11461 

20 

+  0.152  35 

—  0.05068 

—  o.ioi  66 

—  0.11379 

21 

+  0.14586 

—  0.05301  !  —  0.09286     —  0.09760 

21 

+  0.14854 

—  0.05501 

—  0.093  54 

—  0.09658 

22 

+  0.14307 

—  0.05666  i  —  0.08640     —  0.08265 

22 

+  0.14524 

—  0.05833 

—  0.08692 

—  0.081  48 

23 

+  0.14067 

—  0.05951  :  —  0.081  17     —  0.06933 

23 

+  0.14239 

—  0.06086 

—  0.081  53 

—  0.06804 

24 

+  0.13864 

—  0.061  71     —  0.076  93     —  0.057  30 

24 

+  0.13995 

—  0.06278 

—  0.077  17 

—  0.05594 

25 

+  0.13693 

—  0.06338     —  0.07356     —  0.04629 

25 

+  0.13789 

—  0.064  20 

—  0.07368 

—  0.04487 

26 

+  0.13558 

—  0.06464     —  0.07094     —  0.03606 

26 

+  0.13616 

—  O.O6522 

—  0.07096 

—  0.03461 

27 

+  0.13453 

-  0.065  55 

—  0.06898     —  0.02643 

27 

+  0.13478 

—   0.06589 

—  0.06850 

—  0.02497 

28 

+  0.13377 

—  0.066  15 

—  0.06761     —  0.01722 

28 

+  0.133  71 

—   0.06626 

—  0.06744 

—  0.01578 

29 

+  0.13330 

—  0.06649 

—  0.06682     —  0.00827 

29 

+  0.13294 

—   0.06638 

—  0.06655 

—  0.00688 

30 

+  0.133  12 

—  0.06657 

—  0.06655 

+  0.00051 

30 

+  0.13245 

—   0.06626 

—  0.06620 

+  0.001  87 

31 

+  0.13322 

—  0.06641 

—  0.06681     +  0.00930 

31 

+  0.13225 

—   0.06589 

—  0.06636 

+  0.01060 

32 

+  0.13361 

—  0.06600 

—  0.06760     +  0.01823 

32 

+  0.13234 

—  0.055  28 

—  0.06706 

+  0.01944 

33 

+  0.13428 

—  0.065  32 

—  0.06896     +  0.02744 

33 

+  0.132  70 

—  0.06438 

—  0.06831 

+  0.02854 

34 

+  0.13525 

—  0.06433 

—  0.07091     +  0.03706 

34 

+  0.133  34 

—   O.O63  19 

—  0.070  16 

+  0.03803 

35 

+  0.13651 

—  0.06299 

—  0.07353 

+  0.04728 

35 

+  0.13427 

—  O.O6  1  63 

—  0.07264 

+  0.04808 

36 

+  0.138  10 

—  0.061  21 

—  0.07688  :  +  0.05827 

36 

+  0.13549  !  —  0.05963 

—  0.07586 

+  0.05887 

37 

+  0.14001 

—  0.05892 

—  o.cSi  09     +  0.07027 

37 

+  0.13702 

—  0.057  10 

—  0.07991 

+  0.07063 

38 

+  0.14227 

—  0.05597 

—  0.086  30     +  0.083  55 

38 

+  0.13886 

—  0.05392 

-  0.08495 

+  0.08361 

39      +  0.14490 

—  0.052  19 

—  0.09272 

+  0.09844 

39 

+  0.14103 

—  0.04989 

—  0.091  15 

+  0.098  14 

40      +  0.14795 

—  0.047  35 

—  0.10059 

+  0.11535 

40 

+  0.14356 

—  0.04479 

—  0.09876 

+  0.11461 

41      +  0.151  43 

—  0.041  17     —  0.11028 

+  0.13484 

41 

+  0.14646 

—  0.03834 

—  0.108  13 

+  0.133  55 

42      +  0.15543 

—   O.O33  21        —   O.I22  23 

+  0.15760 

42 

+  0.14978 

—  0.030  10 

—  0.11968 

+  0.15562 

43      +  0.16000 

—   O.02294       —   O.I37O6 

+  0.184  57 

43 

+  0.153  57 

—  0.01953 

—  0.13403 

+  0.181  71 

44      +  0.165  26 

—   0.00960 

—  0.15566 

+  0.21700      44 

+  0.15789 

—  0.00590 

—  0.15199 

+  0.21301 

4S      +  O.I7I35 

+   0.00785 

—  0.17920 

+  0.25665 

45 

+  0.16286 

+  O.OII  84 

—  0.17471 

+  0.251  18 

46      +  0.17854 

+   0.03087       —   0.20941 

+  0.30597 

46 

+  0.16867 

+  0.035  14 

—  0.20381 

+  0.29854 

47      +  0.18724 

+  0.061  51      —  0.24875 

+  0.368  55 

47 

+  0.17561 

+  0.06601 

—  0.241  63 

+  0.35846 

48 

+  0.198  20 

+  0.10269     —  0.30091 

+  0.44999 

48 

+  0.18428 

+  0.10735 

—  0.291  64 

+  0.43590 

49 

+  0.21280 

+  0.15866    —  0.37145 

+  0.55751 

49 

+  0.195  77 

+  0.16331 

—  0.35907 

+  0.53846 

50 

+  0.23370 

+  0.235  44     —  0469  14 

+  0.70484 

50 

+  0.21225 

+  0.23985 

—  0.452  1  1 

+  0.67805 

51 

+  0.26643 

+  0.341  67     —  0.608  10 

+  0.91263 

Si 

+  0.23832 

+  0.34549 

-  0.583  82 

+  0.87409 

52 

+  0.32291 

+  0.48903    —  0.81195 

+  1.21644    i  52 

+  0.28404 

+  0.491  88 

-  0.77591 

+  1-15930 

53 

+  0.43039 

+    0.691  21 

—  i.  121  59 

+  1.67867 

53 

+  0.37252 

+  0.69304 

—  1.06556 

+  1-59084 

54 

+  0.65464 

+   0.95589 

—  1.61052 

+  241220 

54 

+  0.55982 

+  0.95881 

—  1.51864 

+  2.271  42 

55 

+  1.16322 

+    L25I  I? 

—  2.414.40 

+  3.621  88 

55 

+  0.98862 

+  1.26617 

-  2.254  78 

+  3.38575 

56 

+  2.39922 

+    1.385  II 

-  3.784  32 

+  5-65379 

56 

+  2.03482 

h  145600 

—  349084 

+  5-25163 

57 

+  5.5H93 

+   0.63808 

—  6.15001 

+  8.90000      57 

+  4.67267 

+  0.91643 

-  5-58909 

+  8.243  78 

58 

+  12.93286 

—   2.92359 

—  10.00926 

+  12.77721      58 

+  11.00705 

—  2.04409 

—  8.06205 

+11.98839 

59 

+26.185  67 

—  11.259  17 

—14.92649 

+  12.46123      59 

+22.761  17 

-  9-475  57 

—13.28560 

+12.554  18 

62 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE  VII.— Continued. 
VALUES  OF  A,  B,  C,  AND  D  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


System  2. 

System  3. 

f 

A 

B 

C 

D 

I 

A 

B 

C 

D 

o 

+29.624  12 

—15.17831 

—1444582 

+  3.98350 

O 

+28.095  38 

—14.29079 

—13.80459 

+  3-72368 

I 

+26.178  17 

—12.85822 

—13-31998 

-  7-55334 

I 

+25.IO8  12 

—12.24065 

—12.86746 

—  6.97990 

2 

+15.05897 

—  5-37464 

-  9.68431 

—  11.06821 

2 

+  14.804  22 

-  5-33468 

—  9.46955 

—1047031 

3 

+  7.08250 

—  0.82744 

—  6.25507 

—  8.775  53 

3 

+   7.I427I 

—  0.97149 

—   6.I7I  21 

—  8.464  1  1 

4 

+  3.272  10 

+  0.673  17 

—  3.94529 

-  5-88003 

4 

+    3.36804 

+  0.54538 

-    3-9I34I 

—  5.74732 

5 

+  1.63783 

+  0.90532 

—  2.543  15 

—  3.84260 

5 

+    1.70907 

+  0.82100    —  2.53007 

-  3-78667 

6 

+  0.92681 

+  0.77463 

—  I-70I45 

—  2.571  56 

6 

+   0.97332 

+  0.721  94 

—    1.69525 

—  2.546  14 

7 

+  0.59776 

+  0.58640 

—  1.18416 

—  1.78590 

7 

+  0.627  53 

+  0.553  13 

—  1.180  66 

—  1.77284 

8 

+  043283 

+  0.42237 

—  0.855  20 

—  1.28747 

8 

+  0452  10 

+  0.40075 

-  0.85285 

—  1.27974 

9 

+  0.34273 

+  0.295  72 

—  0.63846 

—  0.95970 

9 

+  0.35545 

+   O.28l  22 

—  0.63665 

—  0.95445 

10 

+  0.28909 

+  0.201  73 

—  040082 

—  0.73606 

10 

+  0.29760 

+   O.I9I  71 

—  048933 

—  0.732  10 

ii 

+  0.25443 

+  0.13276 

—  0.38720 

-  0.578  14 

ii 

+  0.26020 

+   0.12569 

-  0.38589 

—  0.57491 

12 

+  0.23036 

+  0.082  14 

—  0.31252 

—   0.463  12 

12 

+  0.23428 

+   0.07706       —   0.3II33 

—  0.4^36 

13 

+  0.21260 

+  0.04482 

—  0.25741 

—  0.37700 

13 

+   O.2I5  22 

+    0.041   IO    :     —    0.25634 

—  0.37458 

14 

+  0.19884 

+  0.01709 

—  0.21594 

—   0.31093 

14 

+   0.20055 

+    0.01437 

—  0.21494 

—  0.30876 

J| 

+  0.18778 

—  0.00364 

—  0.184  16 

—   O.259  12 

15 

+   0.18883 

—    O.OO5  63 

—  0.18321 

—  0.257  16 

16 

+  0.17866 

—  0.01925 

—  0.15942 

—   0.21769 

16 

+   O.I792I 

—    O.02O68        —    0.15852 

—  0.21591 

17 

+  0.17099 

—  0.031  07 

—  0.13991 

—   0.18397 

17 

+   O.I7I  14 

—    O.O32O8         —    O.I3906    I    —    0.18235 

18 

+  0.16444 

—  0.04006 

—  0.12437     —  0.15606 

18 

+   0.16429 

—  0.04074      —  0.12355 

—  0.15459 

iQ 

+  0.15880 

—  0.04693 

—  0.11187 

—   O.I3262 

19 

+   O.I584I 

—  0.04733 

—  O.I  1  1  08 

—  0.131  28 

20 

+  0.15392 

—  0.052  17 

—  o.ioi  74 

—   O.II264 

20 

+   0.15333 

—  0.05236 

—  0.10098 

—  0.11143 

21 

+  0.14968 

—  0.056  17  i  —  0.093  50  1  —  0.095  37 

21 

+   0.14894 

—  0.056  1  8 

—  0.092  76 

—  0.09427 

22 

+  0.14600 

—  0.05921 

—  0.08679     —  0.08024 

22 

+   0.145  14 

—  0.05907 

—  0.08606 

—  0.07925 

23 

+  0.14282 

—  0.06  1  50 

—  0.081  32 

—   0.06680 

23 

+   O.I4I  84 

—  0.06  1  24 

—  0.08060 

—  0.06593 

24 

+  0.14009 

—  0.06320 

—  0.07689 

—   0.05470 

24 

+    O.I390I 

—  0.062  82 

—  0.076  17 

—  0.05304 

25 

+  0.13775 

—  0.06441 

—  0.07334 

—   O.O4366 

25 

+   0.13659 

—  0.06395 

—  0.07263 

—  O.O.J300 

26 

+  0.13579 

—  0.06524 

—  0.07055 

-  0.03345 

26 

+   0.13454 

—  0.06470 

—  0.06985 

—  0.03289 

27 

+  0.134  17 

-  0.065  74 

—  0.06844 

—  0.02386 

27 

+   0.13286 

—  0.065  13 

—  0.06773 

—  0.02341 

28 

+  0.13288 

—  0.06595 

—  0.06692 

—  0.01473 

28 

+  0.131  50 

—  0.065  29 

—  0.06622 

—  0.01439 

29 

+  0.131  89 

—  0.06592 

—  0.06598 

—  0.00591 

29 

+  0.13046 

—   O.O65  21 

—  0.06526 

—  000568 

30 

+  0.131  21 

—  0.06564 

—  0.06556 

+  0.00275 

30 

+  0.12972 

—  0.06489 

—  0.06483 

—  0.00287 

31 

+  0.13081 

—  0.065  13 

—  0.06567 

+  0.01  1  37 

31 

+  0.12927 

—  0.06434 

—  0.064  92 

+  o.oi  i  38 

32 

+  0.13068 

—  0.06438 

—  0.06631 

+  0.02009 

32 

+  0.129  10 

—  0.06357 

—  0.065  54 

+  0.01998 

33 

+  0.13084 

—  0.06336 

—  0.06749 

+  0.02905 

33 

+  0.12921 

—  0.06253 

—  0.06669 

+  0.02880 

34 

+  0.131  28 

—  0.06203 

—  0.06925 

+  0.03838 

34 

+  0.12960 

—  0.061  19 

—  0.06842 

+  0.03800 

35 

+  0.13199 

—  0.06035 

—  0.071  65 

+  0.04825 

35 

+  0.13028 

—  0.05950 

—  0.070  77 

+  0.04771 

36 

+  0.13299 

—  0.05823 

—  0.07476 

+  0.05883 

36 

+  0.131  23 

—  0.05740 

—  0.07384 

+  0.058  14 

37 

+  0.13428 

—  0.05558 

—  0.07869 

+  0.07035 

37 

+  0.13248 

—  0.05478 

—  0.077  7i 

+  o.o  '19  47 

38 

+  0.13586 

—  0.05228 

-  0.08359 

+  0.08304 

38 

+  0.13404 

—  0.051  51 

—  0.08252 

+  0.081  97 

39 

+  0.13775 

—  0.048  15 

—  0.08962 

+  0.09723 

39 

+  0.13590 

—  0.04744 

—  0.08846 

+  0.09594 

40 

+  0.13998 

-  0.04295 

—  0.097  02 

+  0.11330 

40 

+  0.138  10 

—  0.04235 

—  0.095  76 

+  0.11177 

41 

+  0.14255 

—  0.03642 

—  0.106  13 

+  0.131  75 

4i 

+   O.T4066 

—  0.03593 

—  0.10473 

+  0.12993 

42 

+  0.14550 

—   0.028  12 

-  0.11737 

+  0.15323 

42 

+   O.I4360 

—  0.02781 

—  0.11580 

+  0.15108 

43 

+  0.14885 

—   0.01755 

—  0.13131 

+  0.17857 

43 

+   0.14658 

—  0.01746 

—  0.12953 

+  0.17604 

44 

+  0.15269 

—  0.00395 

—  0.14875 

+  0.20895 

44 

+   0.15085 

—  0.004  19 

—  0.14668 

+  0.20595 

45 

+  0.15709 

+  0.013  68 

—  0.17077 

+  0.24592 

45 

+   0.15533 

+  0.01300 

-  0.16833 

+  0.24236 

46 

+  0.16222 

+  0.03671 

—  0.19894 

+  0.291  73 

46 

+   0.16059 

+  0.03542 

—  0.10601 

+  0.28747 

47 

+  0.16834 

+  0.067  12 

—  0.23547 

+  0.34957 

47 

+   O.I66O2 

+  0.06406 

—  0.231  89 

+  0.34443 

48 

+  0.17598 

+  0.10768 

—  0.28365 

+  0.424  17 

48 

+   0.17489 

+  0.10427 

—  0.279  15 

+  0.41786 

49 

+  0.186  10 

+  0.16234 

—  0.34843 

+  0.52271 

49 

+   0.18554 

+  0.15709 

—  0.34264 

+  0.51483 

50 

+  0.20072 

+  0.23676 

-  0.43747 

+  0.65644 

50 

+   0.201  00 

+  0.22874 

—  0.42976 

+  0.64633 

Si 

+  0.22403 

+  0.33893 

—  0.56296 

+  0.84360 

51 

+   0.22568 

+  0.32664 

—  0.55232 

+  0.830  17 

52 

+  0.265  29 

+  0.47966 

—  0.74495 

+  1.11472 

52 

+   O.269  1  1 

+  0.46054 

—  0.72966 

+  1.00602 

53 

+  0.34571 

+  0.671  72 

—  1.01742 

+  1.52277 

53 

+   0-35301 

+  0.641  40 

—  0.99442 

+  1.49508 

54 

+  0.51642 

+  0.92341 

—  143982 

+  2.161  97 

54 

+   0.52902 

+  0.87436 

—  1-40338 

+  2.11762 

55 

+  0.90636 

+  1.21194 

—  2.11829 

+  3-20054 

55 

+   0.02556 

+  1-13169 

—  2.05725 

+  3-12267 

56 

+  1.85077 

+  1.39039 

—  3.241  16 

+  4-91924 

56 

+    1.87094 

+    1.262  21 

—  3-133  16 

+  4-76908 

57 

+  4.20451 

+  0.91034 

—  5-11484 

+  7-64885 

57 

+  4-183  37 

+   0.73285 

—  4-91623 

+  7-34183 

58 

+  979293 

-  1-71451 

-  8.07843 

+  11.07242 

58 

+  9.55247 

—    1.83400 

-  7-71848 

+  1047949 

59 

+20.207  75 

—  8.331  97 

-11.87575 

+  11.86403 

59 

+  19-331  99 

-  8.023  85 

—11.308  13 

+  11.06309 

COEFFICIENTS   FOR  DIRECT  ACTION. 


TABLE   VII.—  Continued. 
VALUES  OB-  A,  B,  C,  AND  D  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


System  4. 

System  5. 

i 

A 

B 

C 

D 

f 

A 

B 

C 

D 

0 

+27-753  75 

—13.94094 

—13.81281 

+  2.71293 

0 

+27.63990 

—13.80676 

-13.833  14 

+  1.162  19 

i 

+24.10807 

—11.43147 

—12.67662 

—  7-27193 

i 

+22.709  15 

—10.43790 

—12.27123 

—  8.00809 

2 

+  M.OgO  02 

—  4-81177 

—  9.22826 

—10.18594 

2 

+12.86967 

—  4.11827 

—  8.75142 

—10.01752 

3 

4-    6.82696 

-  0.83885 

-  5.988  13 

—  8.14397 

3 

+  6.17336 

—  0.53043 

—  5.64295 

-   7.784  12 

4 

+    3.24571 

+  0.55049 

—  3.79620 

—  5.53629 

4 

+  2.93411 

+  0.64851 

-  3.58261 

—   5.261  65 

5 

+    I.6593I 

+  0.79841 

-  2.45770 

—  3-661  70 

5 

+  1.50482 

+  0.82485 

—  2.32966 

-   3.48648 

6 

+   0.94972 

+  0.700  16 

—  1.64989 

—  2471  69 

6 

+  0.86547 

+  0.70639 

—  1.57185 

—   2.36290 

7 

+   0.6l3gO 

+  0.53734 

—  1.15126 

—  1.72669 

7 

+  0.56289 

+  0.53922 

—    1.  102  12 

—    I.658o8 

8 

+   0.442  70 

+  0.39037 

—  0.83307 

—  1.24977 

8 

+  0.40883 

+  0.392  14 

—   0.80098 

—  1.205  16 

9 

+  0.348  10 

+  0.27476 

—  0.62286 

-  0.934  13 

9 

+  0.32390 

+  0.27725 

—  0.601  14 

—  0.90424 

10 

+  0.291  44 

+  0.18792 

—  0.47937 

—  0.71780 

10 

+  0.273  1  6 

+  0.191  03 

—  0.464  1  8 

—  0.69723 

ii 

+  0.25483 

+  0.12366 

-  0.37848 

—  0.56454 

ii 

+  0.24042 

+   O.I27  12 

—  0.36755 

—  0.55008 

12 

+  0.22948 

+  0.076  19 

—  0.30568 

—  045265 

12 

+  0.21778 

+  0.07980 

—  0.29759 

—  0.44233 

13 

-j-  0.21089 

+  0.041  01 

—  0.25191 

-  0.368  74 

13 

+  0.201  16 

4-  0.04461 

—  0.245  78 

—  0.361  30 

14 

+  0.19660 

+  0.014  79 

—  0.211  39 

—  0.30429 

14 

+  0.18836 

4-  0.01828 

—  0.20664 

—  0.29891 

IS 

+   O.lSS  20 

—  0.00489 

—  0.18031 

—  0.25370 

15 

+  0.178  13 

—  o.oo  i  56 

—  0.17656 

—  0.24982 

16 

+   0.175  85 

—  0.01974 

—   O.I56  12 

—  0.21322 

16 

+  0.16973 

—  0.01661 

—  0.153  10 

—  0.21047 

17 

+   O.I68O3 

—  0.031  01 

—   0.13703 

—  0.18025 

17 

+  0.16267 

—  0.028  1  1 

—  0.13457 

—  0.17836 

18 

+  0.161  41 

—  0.03959 

—   O.I2I  80 

—  0.15297 

18 

+  0.15668 

—  0.03692 

—  0.11977 

—  0.151  73 

19 

+  0.15571 

—  0.046  15 

—   0.10956 

—  0.13005 

19 

+  0.151  53 

—  0.04369 

—  0.10785 

—  0.12933 

20 

+  0.15082 

—  0.051  17 

—   0.09964 

—  0.11051 

20 

+  0.147  10 

—  0.04892 

—  0.098  19 

—   O.IIO20 

21 

+  0.14657 

—  0.05501 

—   O.O9I  56       —   O.O93  62 

21 

+  0.14328 

—  0.05295 

—  0.09031 

—  0.09364 

22 

+  0.14290 

—  0.05792 

—   0.08498       —   0.07883 

22 

+  0.13997 

—  0.05607 

—  0.08390 

—   O.O79  12 

23 

+  0.13975 

—   O.O6O  12 

—   0.079  62       —   0.065  70 

23 

+  0.137  13 

—  0.05847 

—  0.07868 

—  0.06621 

24 

+  0.13703 

—  0.06  1  75 

—    0.07528       —   0.05388 

24 

+  0.13473 

—  0.06028 

—  0.07444 

—  0.05457 

25 

+  0.13473 

—  0.06293 

—  0.071  80     —  0.043  10 

25 

+  0.13270 

—  0.06  1  64 

—  0.071  06 

—  0.04394 

26 

+  0.13280 

—  0.06374 

—   0.069  07       —   0.033  13 

26 

+  0.131  03 

—  0.062  62 

—  0.06842 

—  0.03409 

27 

+   O.I3I  22 

—  0.06423 

—   0.06700    !    —   0.02377 

27 

+  0.12970 

—  0.06328 

—  0.06641 

—  0.02484 

28 

+   0.12997 

—  0.06445 

—   0.065  51        —   0.01486 

28 

+  0.12869 

—  0.06369 

—  0.06500 

—  0.01602 

29 

+   O.I29O4 

—  0.06444 

—  0.064  59     —  0.006  25 

29 

+  0.12798 

—  0.06385 

—  0.064  13 

—  0.00748 

30 

+   0.12840 

—  0.06421 

—   0.064  19   !    +   O.O02  19 

30 

+  0.12758 

—  0.06379 

—  o.o53  79 

+  0.00092 

3i 

+   O.I28o6 

—  0.063  75 

—  0.06430     +  0.01061 

31 

+  0.12747 

—  0.06351 

—  0.06396 

+  0.00930 

32 

+  0.12799 

—  0.06306 

—  o.  6494  .  4-  0.01912 

32 

+  0.12764 

—  0.06300 

—  0.06465 

+  0.01779 

33 

+  0.12822 

—   O.O62  12 

—  0.066  ii    4-  0.027  87 

33 

+   0.128  12 

—  0.06225 

—  0.06587 

+  0.02652 

34 

+  0.12875 

—   0.06089 

—  0.06784    4-  0.03698 

34 

4-  0.12890 

—   O.O6  1  22 

—  0.06768 

+  0.03565 

35 

+  0.12955 

—  0.05933 

—  0.07022     +  0.04663 

35 

+  0.12998 

—   0.05987 

—  0.070  1  1 

+  0.04533 

36 

+  0.13066 

—  0.05736 

—  0.07329    4-  0.05699 

36 

+  0.131  39 

—   O.058  12 

—  0.07326 

+  0.055  75 

37 

+  0.13207 

—  0.05490 

—  0.077  18     +  0.068  27 

37 

+  0.133  13 

—   0.05589 

—  0.07723 

+  0.067  ii 

38 

+  0.13381 

—  0.05  1  82 

—  0.082  oo     4-  0.080  73 

38 

+  0.135  23 

—   0.05305 

—  0.082  17 

+  0.07968 

39 

+  0.13590 

—  0.04794 

—  0.08795    4-  0.09466 

39 

+  0.13772 

—  0.04945 

—  0.08825 

+  0.09378 

40 

+  0.13834 

—  0.04308 

—  0.09527     +  o.i  10  47 

40 

+  0.14062 

—  0.04489 

—  0.09573 

+  0.10981 

41 

+   O.I4I  20 

—  0.03693 

—  0.10426     +  0.12864 

4i 

+  0.14400 

—  0.03907 

—  0.10493 

+  0.12828 

42 

+   0.14449 

—  0.029  13 

—  0.11536     +  0.14982 

42 

+  0.14793 

—  0.03163 

—  0.11628 

4-  0.14986 

43 

+   0.14829 

—  0.019  17 

—  0.129  14    4-  0.17486 

43 

+  0.15247 

—  0.02208 

—  0.13040 

+  0.17544 

44 

+   O.I527I 

—  0.00634 

—  0.14636    4-  0.20492 

44 

+  0.15779 

—  0.00972 

—  0.14808 

+  0.20620 

45 

+   O.IS785 

+  0.01028 

—  0.168  13 

+  0.241  57 

45 

+  0.16407 

+  0.00638 

—  0.17044 

+  0.24382 

46 

+   0.16399 

4-  0.032  oo 

—  0.19597    +  0.28704 

46 

+  0.171  62 

+  0.02748 

—  0.199  ii 

+  0.29063 

47 

+   O.I?!  48 

4-  0.06063 

—  0.23209    +  0.34455 

47 

+  0.18098 

+  0.05539 

—  0.23637 

+  0.34998 

48 

+   O.lSl  02 

+  0.09873 

—  0.27974    +  041883 

48 

+  0.19304 

+  0.092  6  1 

—  0.28564 

+  042686 

49 

+   O.I939I 

+  0.14990 

—  0.34380    +  0.51708 

49 

+  0.20945 

+  0.14264 

—  0.35207 

+  0.52885 

So 

+   O.2I265 

+  0.219  19 

-  0.431  84     +  0.650  55 

50 

+  0.23330 

+  0.21035 

—  0.44366 

+  0.66781 

SI 

4-  0.24236 

4-  0.31354 

—  0.55590     +  0.83740 

51 

+  0.27084 

+  0.30233 

—  0.573  16 

+  0.86291 

52 

+  0.29393 

+  0.441  78 

—  0.73571      +  1.10789 

52 

+  0.33507 

+  042652 

—  0.761  61 

+  1.146  ii 

53 

+  0.391  71 

+  0.61299 

—  1.00472     +  1.51406 

53 

+  0.45469 

+  0.59001 

—  1.04470 

+  1-572  14 

54 

+  0.59284 

+  0.82829 

—    I.42I  12    :    +   2.I47O2 

54 

+  0.69605 

+  0.78884 

—  1.48488 

+  2.23629 

55 

+  1.03724 

+  1.051  ii 

—   2.08836       +   3.16503 

55 

+  I.2I939 

+  0.97368 

—  2.19308 

+   3.301  12 

56 

+  2.07620 

4-  1.11203 

—   3.18823       +   4.81735 

56 

+  241926 

+  0.94390 

—  3-363  18 

+    5.01076 

57 

+  4.561  ii 

+  0.45028 

-    5.0II40       +    7-34432 

57 

+  5-21973 

+  0.077  57 

—  5.29730 

+  7-54759 

58 

+10.15489 

—  2.290  72 

—  7.864  16     +  10.250  29 

58 

+  11.281  01 

—  3.00564 

-  8.275  35 

+  10.19086 

59 

+19.85615 

-  8.398  13 

—  11.45803     4-10.29566 

59 

+21.04269 

—  9-19381 

—11.84889 

+  9-39027 

64  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

TABLE  VII.— Continued. 
VALUES  OF  A,  B,  C,  AND  D  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


System  6. 

System  7. 

i 

A 

B 

C 

D 

i 

A 

B 

C 

D 

0 

+27.16805 

—13.66015 

-13.50787 

—  0.614  53 

o 

+26.70986 

—13-59511 

—13.11476 

—  2.251  50 

i 

+20.933  58 

—  9.36452 

—11.56906 

—  8.839  67 

i 

+19.44882 

—  846686 

—10.981  95 

—  9.68240 

2 

+11.41364 

—  3.29223 

—  8.121  39 

—  9.88868 

2 

+10.20337 

—  2.547  15 

—  7.65624 

—  9.92208 

3 

+   5.381  20 

—   0.147  12 

—  5-23409 

—  744099 

3 

+  4.702  14 

+  0.24741 

—  4-94955 

—  7.24930 

4 

+   2.541  06 

+   0.80284 

-  3.34388 

—  4.99086 

4 

+  2.18926 

+  0.993  76 

—  3.18300 

—  4-81724 

5 

+    I-30I93 

+   0.88985 

—  2.191  79 

—  3-31041 

S 

+  1.11288 

+  0.98787 

—  2.10076 

—  3-191  27 

6 

+  0.751  59 

+  0.738  57 

—  M90I5 

—  2.25264   '•    6 

+  0.64227 

+  0.79477 

—  1.43702 

—  2.17589 

7 

+  0.49304 

+  0.55884 

—  1.05187 

—  1.58825       7 

+  042478 

+  0.59479 

—  1.01958 

—  1-53905 

8 

+  0.36249 

+  040635 

—  0.76885 

-  I.I5995       8 

+  0.31691 

+  0.43149 

—  0.74840 

—  1.12806 

9 

+  0.291  15 

+  0.28868 

—  0.57982 

—  0.87426        9 

+  0.25897 

+  0.30739 

-  0.56637 

—  0.85330 

10 

+  0.24883 

+  0.20073 

—  0.449  54 

—  0.67695      10 

+  0.225  10 

+  0.21527 

—  0.44039 

—  0.66303 

ii 

+   0.221  64 

+  0.13556 

—  0.357  19 

—  0.53617      II 

+  0.20354 

+  0.14721 

—  0.35075  ;  —  0.52689 

12 

+   0.20285 

+  0.08721 

—  0.29006 

—   043271         12 

+  0.18867 

+  0.09673 

—  0.28540 

—  042657 

13 

+   O.I89O2 

+  0.051  15 

—  0.240  17 

—   0.35465         13 

+  0.17768 

+  0.05904 

—  0.23672 

—  0.35068 

14 

+   0.17833 

+  0.02406 

—  0.202  37 

—  0.24935      14 

+  0.169  10 

+  0.03066 

-  0.19977 

—  0.291  91 

IS 

+   O.I0972 

+  0.003  54 

—  0.17325 

—  0.24679      15 

+  0.162  13 

+  O.O09  12 

—  0.171  25 

—  0.24543 

16 

+   0.16259 

—  O.OI2  IO 

—  0.15049 

—  0.20855      16 

+  0.15630 

—   0.00738 

—  0.14893  i  —  0.20799 

17 

+   0.15658 

—  O.O24  12 

—  0.13248 

—  0.17728      17 

+  0.151  34 

—  O.02O  IO 

—  0.131  25 

—  0.17729 

18 

+   O.I5I46 

—  0.033  39 

—  0.11807 

—  0.151  29 

18 

+  0.14708 

—   O.O2998 

—  0.11709 

—  0.151  73 

19 

+   0.14705 

—  0.04059 

—  0.10646 

—  0.12937 

19 

+  0.14338 

—   O.0377I        —   O.IO568 

—  0.130  i^ 

20 

+   0.14324 

—  0.04621 

—  0.09703 

—  0.11061 

20 

+  0.140  19 

—   O.04378       —   0.09642 

—  O.I  1  1  60 

21 

+   0.13995 

—  0.05060 

—  0.08936 

—  0.09435 

21 

+  0.13745 

—   0.04856       —   0.08887 

—  0.09551 

22 

+  0.137  13 

—  0.05404 

—  0.083  10 

—  0.08005 

22 

+  0.13509 

—   O.05236       —   O.08273 

—  0.081  34 

23 

+  0.13472 

—  0.05672 

—  0.07800 

—  0.06731 

23 

+  0.133  II 

-  0.055  37     -  0.077  73 

—  0.06868 

24 

+  0.13270 

—  0.05881 

—  0.07388 

—  0.05581 

24 

+  O.I3I47 

—  0.05775 

—  0.073  71 

—  0.05724 

25 

+  0.131  04 

—  0.06043 

—  0.07060 

—  0.04528 

25 

+  0.130  15 

—  0.05964 

—  0.07051 

—  0.04674 

26 

+  0.12970 

—  0.061  66 

—  0.06804 

—  0.03551 

26 

+  0.129  14 

—  0.061  13 

—  0.06803    —  0.03698 

27 

+  0.12869 

—  0.06257 

—   O.066  12 

—  0.02631 

27 

+  0.12844 

—  0.062  27 

—  o.o65  19     —  0.027  77 

28 

+  0.12799 

—  0.063  20 

—  O.O6478 

—  0.01752 

28 

+  0.12804 

—  0.063  12 

—  0.06493     —  0.01895 

29 

+  0.127  58 

—  0.06359 

—   O.O64OO 

—  0.00899 

29 

+  0.12793 

—  0.063  72 

—  0.06421     —  0.01037 

30 

+  0.12747 

—  0.06375 

—  0.063  73 

—  0.00059 

30 

+  0.128  II 

—  0.06408 

—  0.06402     —  0.00191 

31 

+  0.12766 

—  0.06368 

—  0.06397 

+  0.00782 

31 

+  0.12853 

—  0.06423 

—  0.06436     +  0.00658 

32 

+  0.128  14 

—  0.06340 

—  0.06475 

+  0.01635 

32 

+  0.12937 

—  0.064  14 

—  0.06523     +  0.01522 

33 

+  0.12893 

—  0.06287 

—  0.06606 

+  0.025  1  6 

33 

+  0.13046 

—  0.06382 

—  0.066  66  ,  +  0.024  IS 

34 

+  0.13003 

—  0.06207 

—  0.06796 

+  0.03439 

34 

+  0.13188 

—  0.06320 

—  0.06867     +  0.03352 

35 

+  0.13146 

—  0.06095 

—  0.07052 

+  0.044  19 

35 

+  0.13365 

—  0.06228 

—  0.071  36     +  0.043  SO 

36 

+  0.13325 

—  0.05943 

—  0.07380 

+  0.05476 

36 

+  0.135  77 

—  0.06097 

—  0.07481     +  0.05429 

37 

+  0.13538 

—  0.05746 

—  0.07793 

+  0.06633 

37 

+  0.13831 

—  0.059  19 

—  0.079  12     +  0.066  ii 

38 

+  0.13794 

—  0.05488 

—  0.08305 

+  0.079  15 

38 

+  0.141  28 

—  0.05683 

—  0.08445     +  0.07925 

39 

+  0.14092 

—  0.051  56 

—  0.08935 

+  0.093  57 

39 

+  0.14474 

—  0.05372 

—  0.091  01 

+  0.09404 

40 

+  0.14439 

—  0.04729 

—  0.097  II 

+   O.I  10  00 

40 

+  0.14874 

—  0.04966 

—  0.09908 

+  0.11093 

4i 

+  0.14843 

—  0.041  77 

—  0.10665 

+  0.12898 

41 

+  0.15339 

—  0.04438 

—  0.10300 

+  0.13048 

42 

+  0.153  10 

—  0.03467 

—  0.11844 

+   O.I5I  20 

42 

+  0.15877 

—  0.037  5O 

—  0.121  27 

+  0.15342 

43 

+  0.15855 

—  0.02546 

—  0.13311 

+  O.I776I 

43 

+  0.16505 

—  0.02853 

—  0.13653 

+  0.18072 

44 

+  0.16495 

—  0.01347 

-  0.15149 

+  0.20947 

44 

+  0.17246 

—  0.01677     —  0.15567 

+  0.213  73 

45 

+  0.17257 

+   O.OO222 

—  0.17480 

+  0.24852 

45 

+  0.181  29 

—  0.00133     —  0.17997 

+  0.25426 

46 

+  0.181  82 

+   0.02290 

—  0.20472 

+  0.29723 

46 

+  0.19206 

+   O.OI9  II        —   O.2II  17 

+  0.30493 

47 

+  0.19336     +  0.05033 

—  0.24370 

+  0.359  18 

47 

+  0.205  57 

+   0.04631 

—  0.251  88 

+  0.36951 

48 

+  0.208  36     +  0.087  oo 

-  0.295  37 

+  0.43967 

48 

+  0.223  17 

+  0.082  74 

—  0.30590 

+  0.45358 

49 

+  0.22884    +  0.13637 

—  0.36520 

+  0.54676 

49 

+  0.24721 

+  0.131  82 

—  0.37903 

+  0.56569 

50 

+  0.25860     +  0.20317 

—  0461  77 

+  0.693  12 

50 

+  0.282  1  1 

+  0.198  18 

—  0.48029 

+  0.71923 

5i 

+  0.305  13    +  0.293  60 

—  0.59874 

+  0.89921 

Si 

+  0.33642 

+  0.28773 

—  0.624  15 

+  0.93591 

52 

+  0.38393    +  0.41479 

—  0.798  70 

+  I.I99I3 

52 

+  0.42773 

+  0.40676 

—  0.834  50 

+  1.25184 

53 

+  0.52866    +  0.57144 

—    1.  100  II 

+  1.651  13 

53 

+  0.59408 

+  0.557  85 

—  I.I5I93 

+  1.72856 

54 

+  0.81650 

+  0.75357 

—  1-57007 

+  2.35573 

54 

+  0.92198 

+  0.725  14 

-  1-64713 

+  2471  57 

55 

+  1.43137 

+  0.895  83 

—  2.327  19 

+  348082 

55 

+  1.61638 

+  0.8274') 

-  2.44383 

+  3.65374 

56 

+  2.81766 

+  0.75750 

-  3-575  16 

+  5.26294 

56 

+  3-16615 

+  0.58408 

—  3.75022 

+  5.50398 

57 

+  5.97720 

—  0.36200 

—  5.61520 

+  7.807  13 

57 

+  6.641  59 

—  0.78455 

—  5.85704 

+  8.051  79 

58 

+12.52832 

-  3-86693 

—  8.661  40 

+  10.10754 

58 

+  13.61205 

—  4.70484 

—  8.90720 

+10.018  16 

59 

+22.18774 

—  10.13334 

—12.054  39 

+  8.285  33 

59 

+23.13200 

—11.051  60 

—  12.08040 

+  7-24047 

COEFFICIENTS  FOR  DIRECT  ACTION. 


TABLE   VII.— Continued. 
VALUES  OF  A,  B,  C,  AND  D  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


System 

8. 

System 

9- 

1 

A 

B 

C 

D 

I 

A 

B 

C 

D 

0 

+27.214  80 

—13.93826 

-13.27655 

—  3.50011 

0 

+29406  78 

—14.96887 

—14.43791 

—  4.12709 

I 

+  19.03987 

—  8.02033 

—11.01953 

—10.66509 

i 

+20.202  62 

—  8.251  18 

—11.95144 

—11.84365 

2 

+  9-631  65 

—  1.99026 

—  7.641  36 

—10.302  17 

2 

+  9.90242 

—  1-721  59 

—  8.18082 

—11.10640 

3 

+  4.31063 

+  0.61744 

—  4.92809 

—  7-32955 

3 

+  4.28872 

—  0.911  18 

-  5.19988 

—  7.71982 

4 

+  1.96054 

+  1.20621 

—  3.16675 

—  4-80935 

4 

+  1.89456 

+  140655 

—  3.301  10 

—  4.984  16 

5 

+  0.98006 

+  1.10985 

—  2.08990 

—  3.16793 

5 

+  0.92598 

+  1.23253 

-  2.15852 

—  3.24867 

6 

+  0.561  76 

+  0.86813 

—  1.42989 

—  2.155  37 

6 

+  0.52426 

+  0.94242 

—  146667 

—  2.19556 

7 

+  0.37331 

+  0.64149 

—  1.01479 

—  1-52413 

7 

+  0.348  15 

+  0.68736 

—  1.03553 

—  1.54604 

8 

+  0.28229 

+  046281 

—  0.74509 

—  Li  17  93 

8 

+  0.26537 

+  049196 

—  0.75734 

—  1.13104 

9 

+  0.23465 

+  0.32936 

—  0.56401 

—  0.84669 

9 

+   O.223  12 

+  0.34843 

—  0.571  56 

—  0.855  24 

10 

+  0.20740 

+  0.23126 

—  0.43866 

—  0.65889 

10 

+   0.19944 

+  0.24406 

-  0.44351 

—  0.66492 

ii 

+  0.19028 

+  0.159  18 

—  0.34946 

—  0.52446 

ii 

+   0.18474 

+  0.16797 

—  0.352  70 

—  0.52899 

12 

+  0.17851 

+  0.10591 

—  0.28442 

—  042532 

12 

+   0.17464 

+  0.11205 

—  0.28667 

—  042889 

13 

+  0.16976 

+  0.06621 

—  0.23597 

—  0.35025 

13 

+   O.I67O8 

+  0.07053 

—  0.23761 

—  0.353  18 

14 

+  0.16285 

+  0.03633 

—  0.19920 

—  0.29205 

14 

+  O.l6lO5 

+  0.03940 

—  0.20043 

—  0.29452 

IS 

+  0.157  17 

+  0.01365 

—  0.17081 

—  0.24598 

15 

+   0.15599 

+  0.01580 

—  0.171  81 

—  0.248  10 

16 

+  0.15234 

—  0.003  74 

—  0.14861 

-  0.20880 

16 

+   O.I5I  67 

—  0.00223 

—  0.14943 

—  0.21065 

17 

+  0.148  19 

—  0.017  17 

—  0.13102 

—  0.17830 

17 

+   0.14790 

—  0.016  16 

—  0.131  73 

—  0.17993 

18 

+  0.14458 

—  0.02763 

—  0.11696 

—  0.15286 

18 

+   0.14458 

—  0.02701 

—  0.11760 

—  0.15430 

19 

+  0.14143 

—  0.03583 

—  0.105  61 

—  0.131  33 

19 

+   O.I4I  69 

—  0.03549 

—  0.10622 

—  0.13262 

20 

+  0.13871 

—  0.04229 

—  0.00641 

—  0.11285 

20 

+  0.139  16 

—  0.042  17 

—  0.09701 

—  0.11400 

21 

+  0.13636 

—  0.04743 

—  0.08894 

—  0.09678 

21 

+  0.13698 

—  0.04747 

—  0.08952 

—  0.097  79 

22 

+  0.134  36 

—  0.051  51 

-  0.08285 

—  0.08260 

22 

+   0.135  12 

—  0.051  70 

—  0.08343 

—  0.083  50 

23 

+  0.13268 

—  0.05478 

—  0.07792 

—  0.06993 

23 

+   0.13357 

—  0.05508 

—  0.07850 

—  0.07072 

24 

+  0.131  33 

—  0.05738 

—  0.07394 

—  0.05846 

24 

+   0.13233 

—  0.05779 

—  0.07454 

—  0.059  14 

25 

+  0.13027 

—  0.05948 

—  0.07080 

—  0.04791 

25 

+  0.131  37 

—  0.05996 

—  0.071  41 

—  0.04850 

26 

+  0.12951 

—  0.061  14 

—  0.06837 

—  0.038  10 

26 

+  0.130  70 

—  0.061  69 

—  0.06901 

—  0.03858 

27 

+  0.12904 

—  0.06245 

—  0.06659 

—  0.02882 

27 

+  0.13031 

—  0.06305 

—  0.06725 

—  0.02921 

28 

+  0.12885 

—  0.06346 

—  0.065  39 

—  0.01993 

28 

+  0.13020 

—  0.064  12 

—  0.06608 

—  O.O2O2I 

29 

+  0.12894 

—   0.064  21 

—  0.06475 

—  0.01127 

29 

+  0.13037 

—  0.06490 

—  0.06546 

—  0.01  1  45 

30 

+  0.12933 

—   O.O647I 

—  0.06463 

—  0.00271 

30 

+  0.13083 

—  0.06544 

—  0.06538 

—  0.00279 

31 

+  0.13002 

—   0.06498 

—  0.06504 

+  0.00589 

31 

+  0.131  59 

—  0.065  74 

—  0.06583 

+  0.00591 

32 

+  0.131  01 

—   O.O65  O2 

—  0.06599 

+  0.01465 

32 

+  0.13263 

—  0.06582 

—  0.06683 

+  0.01478 

33 

+  0.13232 

—   0.06482 

—  0.06751 

+  0.02372 

33 

+  0.13401 

—  0.06563 

—  0.06839 

+  0.02397 

34 

+  0.13397 

—   0.06432 

—  0.06963 

+  0.03325 

34 

+  0.135  71 

—  0.065  15 

—  0.07057 

+  0.03362 

35 

+  0.13596 

—   0.06352 

—  0.07244 

+  0.04341 

35 

+  0.137  77 

—  0.06435 

-  0.07343 

+  0.04392 

36 

+  0.13834 

—   0.06233 

—  0.07602 

+  0.05441 

36 

+  0.14022 

—  0.063  15 

—  0.07706 

+  0.05506 

37 

+  0.141  13 

—   O.o6o66 

—  0.08048 

+  0.06647 

37 

-j-  0.14308 

—  0.06  1  47 

—  0.08  1  59 

+  0.06728 

38 

+  0.14440 

—   0.05840 

—  0.08599 

+  0.07989 

38 

+  0.14640 

—  0.05920 

—  0.087  18 

+  0.08087 

39 

+  0.148  16 

—   0.05541 

—  0.09277 

+  0.09501 

39 

+  0.15022 

—  0.056  18 

—  0.09405 

+  0.096  19 

40 

+  0.152  52 

—   0.05145 

—  o.ioi  08 

+  0.11229 

40 

+  0.15464 

—  0.052  18 

—  0.10246 

+  0.11369 

41 

+  0.15756 

—   0.04626 

—   O.I  1  1  29 

+  0.13231 

41 

+  0.15973 

—  0.04694 

—  0.11279 

+  0.13396 

42 

+  0.163  40 

—  0.03947 

—  0.12392 

+  0.15582 

42 

+  0.16561 

—  0.04007 

—  0.125  54 

+  0.15776 

43 

+  0.17020 

—  0.03058 

—  0.13961 

+  0.18384 

43 

+  0.17245 

—  0.03106 

—  0.141  39 

+  0.186  1  1 

44 

+  0.17822 

—  0.01889 

—  0.15931 

+  0.21772 

44 

+  0.18048 

—  0.01921 

—  0.161  26 

+  0.22039 

45 

+  0.18779 

—  0.00348 

—  0.18430 

+  0.25939 

45 

+  0.19004 

—  0.00358 

—  0.18646 

+  0.26254 

46 

+  0.19946 

+  0.01695 

—  0.21640 

+  0.31152 

46 

+  0.201  66 

+  0.017  16 

—  0.21884 

+  0.31527 

47 

+  0.214  10 

+  0.044  19 

—  0.25828 

+  0.37802 

47 

+  0.21618 

+  0.04488 

—  0.261  04 

+  0.38255 

48 

+  0.233  13 

+  0.08075 

—  0.31387 

+  0.46469 

48 

+  0.23409 

+  0.082  10 

—  0.31709 

+  0.47025 

49 

+   0.259  12 

+  0.13004 

—  0.389  15 

+  0.58041 

49 

+  0.26056 

+  0.13243 

-  0.39209 

+  0.58739 

50 

+   0.20674 

+  0.10672 

—  0.49344 

+   0.739  12 

50 

+  0.29746 

+  0.20073 

—  0.40820 

+  0.748  16 

51 

+   0.35509 

+  0.28662 

—  0.641  69 

+   0.06340 

Si 

+  0.35463 

+  0.29326 

—  0.64789 

+  0.975  66 

52 

+   0.45286 

+  0.40576 

—  0.85863 

+    I.2OO97 

52 

+  0.450  50 

+  0.41679 

—  0.86729 

+  1.30862 

53 

+   0.63050 

+  0.55578 

—  1.18620 

+    1.786  19 

53 

+  0.625  31 

+  0.57426 

-  I.I9957 

+  1.81371 

54 

+  0.980  18 

+  0.71766 

-  1.69783 

+  2.55944 

54 

+  0.971  84 

+  0.74881 

—  1.72064 

+  2.60676 

55 

+  1.72092 

+  0.80056 

—  2.52146 

+  3.791  24 

55 

+  1.71305 

+  0.851  55 

-  2.565  51 

+  3-88150 

56 

+  3.37687 

+  0.495  10 

-  3-871  98 

+  5-7i6i9 

56 

+  3-39986 

+  0.56678 

-3-96664 

+  5-903  18 

57 

+  7.09411 

—  1.04890 

—  6.04521 

+  8.33290 

57 

+  7.271  80 

—  1.00883 

—  6.26296 

+  8.721  65 

58 
59 

+14.50438 

+24.307  55 

—  5.33643 
-11-95739 

—  9.16795 
—12.350  17 

+  10.174  19 
+  6.72653 

58 
59 

+15.22832 
+26.10678 

—  5.57039 
—12.841  66 

—  9.65791 
—13.26509 

+  10.830  84 
+  7-I8745 

66 


ACTION   OF  THE   PLANETS  ON  THE  MOON. 


TABLE  Mil.— Concluded. 

VALUES  OF  A,  B,  C,  AND  D  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


System  10. 

System  n. 

i 

A 

B 

C 

D 

i 

A 

B 

C 

D 

0 

+32.96968 

—16.56997 

—16.399  71 

-  3-66844 

0 

+35-831  31 

—17.896  16 

—17.935  12 

—  1.71241 

i 

+22.87840 

-  9-29885 

—13-57954 

—12.86257 

i 

+25.08088 

—10.06956 

—  15.01132 

—12.871  70 

2 

+11.01961 

—  1.89825 

—  9.121  36 

—12.14882 

2 

+12.64960 

—  2.65575 

—   9-99385 

—12.89708 

3 

+  4-64991 

—  1.01842 

-  5.66834 

—  8.32679 

3 

+  5-31222 

—  0.803  15 

—   6.II536 

—  8.902  19 

4 

+  2.00491 

+  1.52568 

-  3.53058 

-  5-29455 

4 

+  2.27438 

+  148038 

-  3-754  76 

—  5.628  18 

5 

+  0.961  50 

+  1-31385 

—  2.275  35 

-  340808 

5 

+  1.08448 

+  1.30802 

—  2.39250 

-  3-59521 

6 

+  0.53767 

+  0.991  88 

-  1-52955 

—  2.28306 

6 

+  0.602  79 

+  0.991  77 

—  1-59455 

—  2.39023 

7 

+  0.35496 

+  0.71640 

—  1.071  36 

—  1.59699 

7 

+  0.39457 

+  0.715  16 

—  1.10973 

—  1.661  64 

8 

+  0.270  14 

+  0.50875 

—  0.77890 

—  1.16257 

8 

+  0.29706 

+  0.50583 

—  0.80290 

—  1.20341 

9 

+  0.22724 

+  0.35797 

—  0.58520 

—  0.875  82 

9 

+  0.24704 

+  0.35397 

—   0.60  1  02 

—  0.90267 

10 

+  0.20327 

+  0.24927 

-  0.45255 

—  0.67894 

10 

+  0.21868 

+  0.24478 

—  0.46347 

—  0.697  18 

ii 

+  0.18839 

+  0.17058 

—  0.35896 

-  0.53889 

ii 

+  0.20085 

+  0.16598 

—  0.36683 

—  0.551  59 

12 

+  0.178  13 

+  0.11307 

—  0.291  19 

—  0.43609 

12 

+  0.18848 

+  0.10859 

—  0.29707 

—  0.44509 

13 

+  0.17040 

+  0.07059 

—  0.241  oo 

—  0.358  52 

13 

+  0.179  19 

+  0.06636 

—  0.245  53 

—  0.36497 

14 

+  0.16420 

+  0.03887 

—  0.20307 

—  0.298  55 

14 

+  0.171  75 

+  0.03492 

-  0.20668 

—  0.303  18 

IS 

+  0.15900 

+  0.01493 

—  0.17392 

—  0.251  17 

IS 

+  0.16557 

+  o.oi  i  29 

-  0.17686 

—  0.25449 

16 

+  0.15452 

—  0.00331 

—  0.151  19 

—  0.21301 

16 

+  0.16028 

—  0.00665 

—  0.153  64 

-  0.21535 

17 

+  0.15059 

—  0.01736 

—  0.13325 

—  0.181  73 

17 

+  0.15570 

—  0.02038 

—  O.I353I 

—  0.18333 

18 

+  0.147  15 

—  0.02823 

—  0.11892 

-  0.15568 

18 

+  0.151  69 

—  0.03098 

—   O.I2O  7O 

—  0.15670 

19 

+  0.144  13 

—  0.03672 

—  0.10739 

—  0.13365 

19 

+  0.148  17 

—   O.O39  22 

—   0.10896 

—  0.13421 

20 

+  0.141  48 

—  0.04340 

—  0.09808 

—  o.i  14  75 

20 

+  0.145  09 

—  0.045  66 

-  0.09945 

—  0.11495 

21 

+  0.139  19 

—  0.04867 

—  0.09050 

—  0.09831 

21 

+  0.14243 

—  0.05070 

—  0.091  72 

—  0.09823 

22 

+  0.13722 

—  0.05286 

—  0.08436 

—  0.08381 

22 

+   0.140  12 

—  0.054(18 

—  0.08545 

—  0.083  50 

23 

4-  0.135  57 

—  0.05620 

—  0.07938 

—  0.07086 

23 

+  0.138  16 

—  0.05781 

—  0.08036 

—  0.07035 

24 

+  0.13422 

—  0.058  85 

-  0.075  38 

—  0.059  13 

24 

+  0.13653 

—  0.060  27 

—  0.07626 

—  0.05846 

25 

+  0.133  17 

—  0.06096 

—  0.07222 

—  0.04836 

25 

+  0.13522 

—   O.O')220 

—  0.07.1  01 

-  0.04755 

26 

+  0.13241 

—  0.06263 

—  0.06979 

—  0.03832 

26 

+  0.13420 

—   0.063  70 

—  0.070  =;o 

—  0.03741 

27 

+  0.131  93 

—  0.06393 

—  0.06801 

—  0.02884 

27 

+  0.13348 

—   0.06483 

—  0.06864 

—  0.02783 

28 

+  0.131  73 

—  0.06492 

—  0.06681 

—  0.01975 

28 

+  0.13305 

—   0.06565 

—  0.06738 

—  0.01866 

29 

+  0.131  82 

—  0.06563 

—  0.066  18 

—  0.01090 

29 

+  0.13289 

—   O.O6620 

-  0.06668 

—  o.oop75 

30 

+  0.132  18 

—  0.066  10 

—  0.06608 

—  O.OO2  15 

30 

+  0.13302 

—   0.05651 

—  0.06651 

—  0.00096 

31 

+  0.13284 

—  0.06632 

—  0.06651 

+  O.OO662 

3i 

+  0.13344 

—   0.06656 

—  0.06687 

+  0.00785 

32 

+  0.13379 

—  0.06631 

—  0.06749 

+  0.01557 

32 

+  0.134  15 

—   O.O6637 

—  0.06777 

+  0.01682 

33 

+  0.13506 

—  0.06603 

—  0.06903 

+  0.02482 

33 

+  0.135  15 

—   0.065  91 

—  0.06923 

+  0.02608 

34 

+  0.13664 

—  0.06545 

—  0.071  18 

+  0.03454 

34 

+  0.13646 

—  0.065  16 

—  0.071  30 

+  0.035  78 

35 

+  0.13857 

—  0.06454 

—  0.07402 

+  0.04489 

35 

+  0.138  10 

—  0.06405 

—  0.07404 

+  0.046  ii 

36 

+  0.14086 

—  0.063  24 

—  0.07763 

+  0.05608 

36 

+  0.14007 

—  0.06254 

—  0.07755 

+  0.05724 

37 

+  0.14355 

—  0.061  43 

—  0.082  13 

+  0.06834 

37 

+  0.14242 

—  0.060  50 

—  0.081  92 

+  0.06942 

38 

+  0.14668 

—  0.05901 

—  0.08767 

+  0.081  06 

38 

+  0.145  15 

—  0.05784 

—  0.08732 

+  0.08293 

39 

+  0.15029 

—  0.05581 

—  0.09447 

+  0.09730 

39 

+  0.14832 

—  0.05437 

—  0.09395 

+  0.098  1  1 

40 

+  0.15444 

-  0.051  63 

—  0.10280 

+  0.11480 

40 

+  0.15196 

—  0.04987 

—  0.10209 

+  0.11540 

41 

+  0.15920 

—  0.046  17 

—  0.11305 

+  0.13504 

41 

+  0.156  14 

—  0.04406 

—  0.11208 

+  0.13536 

42 

+  0.16470 

—  0.03903 

—  0.12568 

+  0.158  78 

42 

+  0.16093 

—  0.03652 

—  0.12442 

+  0.15873 

43 

+  0.171  06 

—  0.02969 

—  0.141  37 

+  0.18703 

43 

+  0.16646 

—  0.02671 

—  0.13973 

+  0.18647 

44 

+  0.17850 

—  0.01745 

—  0.161  05 

+   O.22I  l6 

44 

+  0.17285 

—  0.01391 

—  0.15804 

+  0.21993 

45 

+  0.18729 

—  o.ooi  31 

—  0.18599 

+   0.26306 

45 

+  0.18036 

+  0.00293 

—  0.18328 

+  0.26093 

46 

+  0.19790 

+   O.O2O  12 

—  0.21801 

+   0.31543 

46 

+  0.18932 

+  0.02521 

—  0.214  53 

+  0.31208 

47 

+   O.2II  OS 

+   0.04873 

-  0.25978 

+   0.382  19 

47 

+  0.20031 

+  0.05497 

—  0.255  27 

+  0.377  IS 

48 

+   O.22798 

+   0.08726 

—  0.31523 

+   0.469  15 

48 

+  0.21431 

+  0.09504 

—  0.30935 

+  0.461  76 

49 

+   O.25O86 

+   0.13952 

—  0.39037 

+   0.58526 

49 

+  0.23309 

+  0.14953 

—  0.38262 

+  0.57453 

50 

+   0.283  78 

+   O.2I084 

—  0.49461 

+   0.74461 

50 

+  0.26004 

+  0.22424 

—  0.48)27 

+  0.729  10 

5i 

+  0.33481 

+   0.30836 

—  0.643  17 

+   0.97029 

Si 

+  0.301  96 

+  0.32727 

—  0.62923 

+  0.94779 

52 

+   0.42084 

+   0.44062 

—  0.861  46 

+    I.3OI  21 

52 

+  0.37330 

+  0.46914 

—  0.84244 

+  1.26849 

53 

+   0.57924 

+  0.614  18 

—  I-I9343 

+    I.8(D5  20 

53 

+  0.50661 

+  0.66075 

—  I.I6737 

+  1.75769 

54 

+   0.89775 

+  0.81982 

—  I-7I755 

+  2.602  39 

54 

+  0.77960 

+  0.00268 

—  1.68227 

+  2.53509 

55 

+    I.5929I 

+  0.08393 

—  2.57684 

+  3-90143 

55 

+  1.38811 

+  1.14402 

—  2.532  13 

+  3-81521 

56 

+   3-2I360 

+  0.81477 

—  4.02836 

+  6.01562 

56 

+  2.84329 

+  1.14104 

-  3-98524 

+  5-94655 

57 

+    7.07792 

—  0.596  54 

—  6.481  38 

+  9-12585 

57 

+  6.43787 

+  0.05782 

—  6.495  70 

+  9-252  71 

58 

+  15.48372 

—  5.18993 

—  IO.2Q-58O 

+11.00597 

58 

+  14.71782 

—  4.16263 

—  IO.W20 

+  12.78088 

59 

+28.021  10 

—13.352  48 

-14.66860 

+  8.801  20 

59 

+28.397  74 

—12.84884 

-15.54889 

+10.00673 

COEFFICIENTS  FOR  DIRECT  ACTION. 


67 


TABLE  VIII. 
DEVELOPMENT  OF  A,  B,  C,  AND  D  FOR  VENUS  IN  PERIODIC  SERIES. 


Coeff.  of 
V,  g' 

^ 

1 

? 

< 

f~* 

1 

9 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

o     o 

+2.1947 

o.oooo 

-0.5886 

o.oooo 

—  i  .606  1 

O.OOOO 

—0.0005 

0.0000 

0+   I 

+0.1742 

—0.0428 

—0.0314 

+0.0085 

—0.1428 

+0.0343 

+0.0080 

+0.0774 

O+  2 

+0.0523 

—0.0556 

—0.0086 

+0.0084 

—0.0438 

+0.0471 

+0.0124 

+0.0131 

o+  3 

+0.0046 

—0.0075 

—  0.0008 

+0.0012 

—0.0038 

+0.0063 

+O.OO22 

+0.0016 

o+  4 

+O.OOOI 

—0.0015 

o.oooo 

o.oooo 

—  O.OOOI 

+O.OOI2 

+0.0004 

o.oooo 

o+  5 

—  O.OOOI 

—  O.OOO2 

0.0000 

+O.OOO2 

O.OOOO 

+0.0002 

0.0000 

o.oooo 

o+  6 

0.0000 

o.oooo 

o.oooo 

o.oooo 

O.OOOO 

O.OOOO 

O.OOOO 

o.oooo 

i-  7 

o.oooo 

o.oooo 

—  O.OOOI 

+O.OOOI 

o.oooo 

X  O.OOOO 

+O.OOOI 

0.0000 

—  6 

—0.0004 

+0.0003 

—0.0003 

o.oooo 

+O.OOOI 

—0.0003 

0.0000 

—  O.OOOI 

—  5 

+0.0004 

+0.0014 

—  O.OOO2 

—0.0005 

—  O.OOOI 

—  O.OOIO 

+0.0005 

—0.0005 

—  4 

+0.0049 

+0.0084 

—  0.0013 

—0.0017 

—0.0038 

—0.0068 

+0.0031 

—  O.OO2I 

—  3 

+0.0546 

+0.0552 

—0.0127 

—0.0115 

—0x5423 

—0.0441 

+0.0182 

—0.0199 

—  2 

+0.2534 

+0.0552 

—0.0820 

—0.0156 

—0.1708 

—0.0393 

+O.O2OI 

—0.1214 

—    I 

+4.0006 

—0.0006 

—1.1590 

+0.0004 

—2.8416 

0.0000 

—0.0015 

—  I.I002 

1+   0 

+0.0924 

—0.0288 

+0.01  18 

+0.0030 

—  0.1036 

+0.0255 

—  O.OOIO 

+0.0525 

1+    I 

+0.0501 

—0.0549 

—  0.0058 

+0.0060 

—0.0445 

+0.0490 

+0.0063 

+0.0068 

1+   2 

+0.0038 

—0.0064 

—  O.OOO2 

+0.0008 

—0.0036 

+0.0059 

+0.0013 

+0.0009 

1+3 

O.OOOO 

—  0.0014 

O.OOOO 

+0.0005 

O.OOOO 

+0.0013 

+0.0004 

+O.OOOI 

1+  4 

—0.000  1 

—  O.OOO2 

O.OOOO 

o.oooo 

+O.0002 

+0.0004 

—  O.OOO2 

0.0000 

i+  5 

o.oooo 

o.oooo 

—O.OOOI 

+O.OOOI 

0.0000 

0.0000 

+O.OOOI 

0.0000 

2-  8 

o.oooo 

o.oooo 

—O.OOOI 

—  O.OOOI 

+O.OOOI 

o.oooo 

+O.OOOI 

0.0000 

2—   7 

O.OOOO 

+0.0004 

—  O.OOO2 

—O.OOOI 

+O.OOOI 

—  0.0006 

+O.OOO2 

O.OOOO 

2-   6 

+0.0004 

+0.0015 

O.OOOO 

—0.0005 

—  O.OOO2 

—  O.OOI  I 

+O.OOO2 

—  OXIOOI 

2-   S 

+0.0054 

+0.0092 

—o.oo  1  6 

—0.0025 

—  0.0036 

—0.0065 

+0.0037 

—0.0027 

2—  4 

+0.0567 

+0.0542 

—0.0174 

—0.0144 

—0.0398 

—0.0308 

+0.0223 

—0.0252 

2—  3 

+0.3093 

+0.0648 

—0.1281 

—0.0245 

—0.1826 

—0.0402 

+0.0315 

—0.1729 

2—   2 

+3.7105 

—0.0014 

—1.3562 

+0.00  1  1 

—2.3535 

o.oooo 

—  0.0018 

—1.6716 

2—    I 

+0.0402 

-0.0173 

+0.0321 

+O.OOIO 

—0.0730 

+0.0162 

—00016 

+0.0516 

2+   O 

+0.0483 

—0.0536 

—0.0049 

+0.0052 

—0.0430 

+0.0486 

+O.OOOI 

+O.OOI2 

2+    I 

+0.0032 

—0.0054 

—0.0003 

+0.0003 

—0.0028 

+0.0048 

+0.0008 

+0.0007 

2+   2 

+0.0001 

—  O.OOIO 

o.oooo 

+O.OOO2 

.00000 

+O.OOII 

+O.OOOI 

0.0000 

2+   3 

—  O.0002 

—0.0002 

—0.0003 

—  O.OOOI 

+0.0003 

—  O.OOO2 

+0.0002 

o.oooo 

2+  4 

O.OOOO 

O.OOOO 

—O.OOOI 

—  O.OOOI 

+O.OOOI 

O.OOOO 

+O.OOOI 

0.0000 

3—  9 

—O.OOOI 

+O.OOOI 

—O.OOOI 

O.OOOO 

+O.OOOI 

o.oooo 

+O.OOOI 

+O.OOOI 

3-8 

o.oooo 

+0.0006 

o.oooo 

—0.0003 

+O.OOOI 

—0.0003 

+O.OOOI 

0.0000 

3-  7 

+0.0004 

+O.OO2O 

+0.0001 

—0.0007 

—0.0003 

—  O.OOI  I 

+0.0007 

—  O.OOO2 

3-6 

+0.0062 

+0.0093 

—  O.OO2O 

—0.0032 

—0.0041 

—0.0064 

+0.0046 

—0.0033 

3-  5 

+0.0586 

+0.0524 

—0.0217 

—0.0168 

—0.0366 

—0.0354 

+0.0252 

—0.0305 

3-  4 

+0.3454 

+0.0705 

—0.1657 

—0/5317 

-0.1789 

—0.0387 

+0.0408 

—  0.2154 

3-  3 

+3.3553 

—  0.0017 

-14697 

+O.OOI2 

-1.8858 

+0.0003 

—0.0014 

—1.8630 

3—  2 

+0.0042 

—0.0079 

+0.0456 

—0.0014 

—0.0490 

+0.0004 

+0.0008 

+0.0547 

3—  i 

+0.0454 

—0.0516 

—0.0044 

+0.0049 

—0.0410 

+0.0463 

—0.0056 

—0.0043 

3-  o 

+0.0030 

—0.0045 

—  O.OOO2 

O.OOOO 

—0.0030 

+0.0044 

O.OOOO 

+0.0003 

3—  i 

—0.0003 

—O.OOIO 

+0.0004 

+O.OOO2 

+O.OOOI 

+O.OOII 

+O.OOO2 

OJOOOO 

3—  2 

—0.0004 

O.OOOO 

—0.0002 

—  O.OOOI 

+.OOOO2 

—  O.OOO2 

o.oooo 

+O.OOOI 

3-  3 

—O.OOOI 

+O.OOOI 

—  O.OOOI 

o.oooo 

+O.OOOI 

0.0000 

+O.OOOI 

+O.OOOI 

4—  10 

—  O.OOOI 

+O.OO02 

+O.OOOI 

+O.OOOI 

+O.OOO2 

o.oooo 

+O.OOOI 

0.0000 

4—  9 

—O.OOOI 

+0.0004 

+0.0003 

+O.OOOI 

O.OOOO 

—  O.OOO2 

O.OOOO 

—  O.0002 

4-8 

+0.0003 

+0.0018 

o.oooo 

—0.0008 

—O.OOOI 

—0.0014 

+0.0013 

—  0.0003 

4-  7 

+0.0064 

+0.0096 

—0.0025 

—  0.0036 

—0.0038 

—  0.0060 

+0.0052 

—0.0036 

4-  6 

+0.0584 

+0.0408 

—0.0252 

—  0.0190 

—0.0332 

—0.0308 

+0.0267 

—0.0344 

4-  S 

+0.3605 

+0.0726 

—0.1938 

—0.0370 

—0.1671 

—0.0355 

+0.0468 

—  0.2442 

4—  4 

+2.9597 

—  0.0013 

-1.4783 

+O.OOI2 

-1.4817 

+O.OOO2 

—  0.0014 

—  1.8596 

4—  3 

—0.0206 

—0.0005 

+0.0526 

—0.0043 

—0.0330 

+0.0045 

+0.0037 

+0.0580 

4—  2 

+0.0424 

—0.0483 

—  0.0058 

+0.0059 

—0.0369 

+0.0420 

—  0.0090 

—0.0075 

4—  I 

+0.0023 

—0.0034 

o.oooo 

—  O.OOOI 

—0.0025 

+0.0034 

O.OOOO 

+0.0003 

68 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE   VIII.— Concluded. 
DEVELOPMENT  OF  A,  B,  C,  AND  D  FOR  VENUS  IN  PERIODIC  SERIES. 


Coeff.  of 
V,  g' 

A 

B 

C 

D 

cos 

sin 

COS 

sin 

COS 

sin 

cos 

sin 

4+  o 

—  O.OO02 

—0.0010 

+O.OOOI 

+O.OOOI 

+0.0002 

+0.0013 

—  O.OOO2 

—0.0003 

4+  i 

o.oooo 

+O.OOO2 

—  O.OOOI 

—  O.OOOI 

O.OOOO 

+0.0003 

—  O.OOOI 

0.0000 

4+  2 

+O.OOOI 

+O.OOO2 

—  O.OOOI 

+O.OOOI 

+0.0004 

o.oooo 

+0.0001 

O.OOOO 

s—  ii 

O.OOOO 

o.oooo 

—  O.OOOI 

o.oooo 

+O.OOOI 

—O.OOOI 

O.OOOO 

—O.OOOI 

S-io 

—  0.0004 

o.oooo 

—  O.OOOI 

—0.0004 

+O.OOOJ 

+O.OOOI 

+O.OOO2 

o.oooo 

S-  9 

+0.0004 

+0.0018 

—  O.OOOI 

—0.0008 

O.OOOO 

—  o.ooii 

+0.0011         —0.0003 

5-8 

+0.0063 

+0.0095 

—0.0028 

—0.0036 

—0.0037 

—0.0056 

+0.0057 

—0.0040 

5-  7 

+0.0579 

+0.0465 

—0.0284 

—  O.0200 

—0.0298 

—0.0268 

+0.0276 

—0.0374 

5-6 

+0.3602 

+0.0712 

—  0.2094 

—  0.0400 

—0.1501 

—0.0313 

+0.0497 

—0.2587 

5-  5 

+2.5539 

—0.0014 

—  1.4070 

+O.OOI2 

-1.1467 

+O.OOO2 

—0.0015 

—1.7422 

5—  4 

—0.0350 

+0.0047 

+0.0564 

—0.0068 

—  0.0205 

+0.0016 

+0.0067 

+0.0598 

5—  3 

+0.0389 

—0.0445 

—0.0064 

+0.0069 

—0.0328 

+0.0375 

—  O.OI22 

—  0.0108 

5—  2 

+0.0018 

—  0.0028 

+O.OOOI 

+O.OOOI 

—0.0018 

+0.0029 

+O.OOO2 

+O.0002 

5—  i 

+0.0004 

—  O.OOI  I 

o.oooo 

+O.OOOJ 

+O.OOO2 

+O.OOI2 

—0.0002 

O.OOOO 

5+  o 

—  O.OOOI 

+O.OOO2 

—0.0002 

+O.OOO2 

—  O.OOOI 

+0.000  1 

+O.OO02 

O.OOOO 

5+  I 

o.oooo 

O.OOOO 

—  O.OOOI 

O.OOOO 

+0.0001 

—O.OOOI 

O.OOOO 

—O.OOOI 

6-12 

—  O.OOOI 

o.oooo 

o.oooo 

o.oooo 

+O.OOO2 

o.oooo 

+O.OOOI 

—  O.OOO2 

6-n 

—O.OOOI 

+0.0003 

+O.OOO2 

OJOOOO 

+0.0004 

+O.OOOI 

+O.COO2 

o.oooo 

6—io 

+0.0003 

+0.0018 

—  O.OOO2 

—0.0008 

—0.0005 

—0.0009 

+O.OOI2 

—0.0005 

6-9 

+0.0061 

+0.0091 

—0.0035 

—0.0038 

—0.0033 

—  0.0050 

+0.0060 

—0.0040 

6-  8 

+0.0557 

+0.0428 

—0.0295 

—0.0203 

—  0.0264 

—0.0225 

+0.0274 

—0.0380 

6-  7 

+0.3455 

+0.0684 

—0.2150 

—0.0414 

—0.1312 

—0.0274 

+0.0500 

—0.2590 

6-  6 

+2.1645 

—0.0007 

-1.2859 

+0.0009 

-0.8788 

—  O.OOOI 

—0.0009 

—1.5648 

6-  5 

—0.0436 

+0.0086 

+0.0558 

—0.0084 

—0.0128 

+O.OO02 

+0.0086 

+0.0588 

6-4 

+0.0356 

—0.0403 

—0.0075 

+0.0078 

—0.0282 

+0.0321 

—O.OI39           —  O.OI2O 

6-3 

+0.0016 

—0.0022 

O.OOOO 

—O.OOOI 

—0.0018 

+O.O02I 

+O.OOO2            —  O.OOOI 

6-   2 

O.OOOO 

—0.0009 

+O.0002 

+O.OOO2 

—0.0005 

+0.0009 

o.oooo 

+0.0005 

6-  i 

—  O.OOOI 

+O.OOO2 

—  O.OOO2 

+O.OOOI 

—O.OOOI 

+O.OOOI 

—  O.OOO2 

—  0.0006 

6+  o 

—O.OOOI 

0.0000 

o.oooo 

o.oooo 

—  O.O002 

o.oooo 

—  O.OOOI 

—  O.OOO2 

7—13 

—0.000  1 

0.0000 

O.OOOO 

—  O.OOOI 

—O.OOOI 

o.oooo 

—0.0002 

O.OOOO 

7—12 

o.oooo 

+0.0005 

+0.0003 

—  O.OOOI              —  O.OOOI 

—O.OOOI 

+0.0004 

+0.0002 

7—  ii 

+0.0006 

+0.0017 

+0.0001 

—0.0009             —  O.OOO2 

—0.0009 

+0.0007 

—  O.OOOI 

7—io 

+0.0064 

+0.0088 

—0.0033 

—0.0042         —0.0030 

—  0.0046 

+0.0058 

—0.0043 

7—  9 

+0.0532 

+0.0392 

—0.0305 

—  O.O2OO             —  O.0223 

—0.0184 

+0.0260 

—0.0386 

7-8 

+0.3227 

+0.0632 

—  O.2IIJ 

—0x1407         —0.1116 

—  0.0226 

+0.0486 

—0.2508 

7—  7 

+1.8074 

—0.0006 

-1.1384 

+0.0008       —0.6689 

—  O.OOO2 

—  0.0009 

—1.3643 

7-6 

—0.0458 

+0.0106 

—0.0547 

—  O.OIO2             —O.OO74 

—  o.ooog 

+0.0105 

+0.0567 

7—  5 

+0.0315 

—0.0359 

—0.0074 

+0.0082         —0.0240 

+0.0274 

—0.0148 

—0.0129 

7—  4 

+O.OOI2 

—0.0015 

+O.OOOI 

—0.0004              —0.0012 

+0.0016 

+O.OO02 

+O.OOO2 

7—  3 

O.OOOO 

—  O.OOII 

+0.0003 

+0.0003            o.oooo 

+0.0008 

—0.0005 

+O.0002 

7-  2 

—O.OOOI 

—0.0002 

—O.OOOI 

+0.0002                 0.0000 

+0.0001 

—O.OOOI 

O.OOOO 

7-  i 

—O.OOOI 

o.oooo 

o.oooo 

—  O.OOO2              —O.OOOI 

o.oooo 

—  O.OO02 

O.OOOO 

8—14 

—  O.OOOI 

O.OOOO 

0.0000 

—O.OOOI                 O.OOOO 

—  O.OOO2 

+O.COOI 

o.oooo 

8-13 

—  O.OOO2 

+O.OOO2 

+O.OOO2 

—  O.OOO2                  O.OOOO 

—O.OOOI 

+O.OO02 

—0.0003 

8-12 

+O.OO02 

+0.0014 

—0.0002 

—0.0008              —0.0002 

—0.00  10 

+0.0013 

—0.0006 

8-ii 

+0.0058 

+0.0080 

—  0.0034 

—0.0044          —  0.0027 

—0.0037 

+0.0050 

—0.0041 

8—io 

+0.0496 

+0.0351 

—0.0306 

—0.0190         —0.0193 

—0.0155 

+0.0250 

—0.0374 

8-  9 

+0.2941 

+0.0577 

—0.1996 

—0.0385        —0.0942 

—0.0192 

+0.0459 

—0.2335 

8-  8 

+14892 

—O.OOOI 

—0.9842 

+0.0003         —0.5050 

—O.OOOI 

—0.0006 

—1.1634 

8-  7 

—0.0461 

+0.0118 

+0.0500 

—  0.0104          —  0.0048 

—O.OOI  2 

+0.0115 

+0.0508 

8-  6 

+0.0278 

—0.0316 

—0.0081 

+0.0085          —  o.o  ic£ 

+0.0228 

—0.0144 

—  O.OI22 

8-  5 

+O.OOIO 

—0.00  10 

+O.OOOI 

—0.0004              —  0.00  T  2 

+0.0012 

—0.0005 

.0.0000 

8-4 

—O.OOOI 

—0.0009 

o.oooo 

—  O.OOOI              —  O.OOO2 

+0.0009 

+O.OOOI 

+O.OOO2 

8-  3 

+O.OOO2 

o.oooo 

o.oooo 

—  O.OOO2              —O.OOOI 

+O.OO02 

—O.OOOI 

+0.0005 

8-   2 

—  O.OOOI 

o.oooo 

o.oooo 

—O.OOOI                   O.OOOO 

—0.0002 

+O.OOOI 

O.OOOO 

COEFFICIENTS   FOR  DIRECT  ACTION. 


69 


TABLE   IX. 

COMPUTATION  OF  THE  COEFFICIENTS  FOR  THE  HANSENIAN  VENUS-TERM  OF 

LONG  PERIOD. 


System. 

A" 

A, 

A,2 

A, 

A,2 

O 

+33-3948 

—48.0672 

—15.2486 

+  1.6810 

+  1.6959 

I 

+30.4226 

—44.3154 

-14-3087 

+  5-2845 

+  5.6929 

2 

+27.3802 

—  404460 

—13-3384 

+  6.9388 

+  7-7952 

3 

+25.7695 

—3&3437 

-12.7915 

+  6.7824 

+  7.8308 

4 

+25.3715 

—37-7570 

—12.6090 

+  5-0267 

+  5.8930 

5 

+25.2326 

—37-5000 

-12.5055 

+  1-9933 

+  2.3924 

6 

+24.7525 

—36-8595 

—12.3300 

-  14599 

—  1.0609 

7 

+24.2976 

—36.3246 

—  12.2190 

—  4.35i6 

—  5.1130 

8 

+24.8273 

—37-1237 

-12.4835 

—  6.2219 

-  7.2996 

9 

+27.0732 

-40.1587 

—13-3322 

—  6.9149 

—  7.9252 

10 

+30.7124 

—44.8651 

—14.5505 

—  5-9534 

—  6.5819 

II 

+33.6483 

—48.4814 

—154035 

-  2.7899 

—  3-0391 

B»" 

A, 

** 

B.. 

A, 

o 

—19.9136 

+27.3868 

+  7-9770 

—  1-3391 

—  1.2601 

i 

—18.5130 

+25.9046 

+  7.7612 

—  4.0210 

—  4.0255 

2 

—  16.9031 

+24.1227 

+  74655 

—  5-0839 

-  5.2866 

3 

-15.8666 

+22.8730 

+  7.2028 

—  4-7701 

—  5.0902 

4 

—154249 

+22.2499 

+  7-0232 

—  34567 

-  3.7608 

5 

-15.2448 

+21.9597 

+  6.9235 

—  14491 

—  1.6516 

6 

—15.0836 

+21.7873 

+  6.9008 

+  0.8330 

+  O.SIQI 

7 

—15.0404 

+21.8370 

+  6.9698 

+  2.9218 

+  3.1445 

8 

-15-4663 

+22.4592 

+  7-1627 

+  4.4869 

+  4.9005 

9 

-16.6559 

+23.9194 

+  74839 

+  5.2206 

+  5.6178 

10 

—18.4569 

+25.9344 

+  7.8260 

+  4-5771 

+  4.7687 

ii 

—19.9292 

+27.4474 

+  8.0139 

+   2.I2OI 

+  2.1782 

C" 

°0 

c* 

cc« 

Ql 

c.* 

o 

—13.4811 

+20.6804 

+  7.2716 

-   0.3420 

—  0.4358 

i 

—11.9096 

+18.4107 

+  6.5475 

—    1.2636 

—  1.6673 

2 

—10.4771 

+16.3233 

+  5-8729 

-    1.8550 

—  2.0586 

3 

—  9.9030 

+15-4707 

+  5.5887 

—   2.OI22 

-  2.7407 

4 

—  9.9466 

+15-5072 

+  5-5856 

—    1.5700 

-  2.1323 

S 

-  9.9878 

+15.5402 

+  5.5820 

—   0.5442 

—  0.7409 

6 

—  9.6689 

+15.0722 

+  54291 

+   0.627O 

+  0.8508 

7 

—  9.2573 

+14.4877 

+  5-2492 

+    14209 

+  1.9685 

8 

-  9.3611 

+14.6645 

+  5-3207 

+    I-735I 

+  2.3991 

9 

—104173 

+16.2394 

+  5.8483 

+    1.6943 

+  2.3074 

10 

—12.2555 

+18.9306 

+  6.7244 

+    1-3762 

+  1.8132 

ii 

—13.7191 

+21.0340 

+  7.3897 

+  0.6699 

+  0.8608 

A" 

A, 

A, 

A. 

A. 

o 

+  1.1897 

-  1-3484 

—  0.2485 

—29.8546 

—38.9716 

i 

+  3-5868 

—  4-2373 

—  0.8724 

—27.5111 

-36.6653 

2 

+  4.5580 

—  5-5349 

—  1.2117 

—24.9093 

—34.0089 

3 

+  4-3152 

—  5-3345 

—  I-2I53 

—23.3292 

—32.2553 

4 

+  3.1531 

—  3-9457 

—  0.9269 

—22.7393 

—314609 

5 

+  1.3233 

—  1.7080 

—  0.4334 

-22.5117 

—31.0906 

6 

—  0.7706 

+  0.8979 

+  0.1689 

—22.2069 

—30.7822 

7 

—  2.6659 

+   3-3202 

+  0.7653 

—22.0282 

—30.7403 

8 

—  4-0554 

+  5-1025 

+  1.2108 

—22.6267 

-31.5788 

9 

-4-6896 

+  5.8349 

+  1.3576 

—24.5324 

—33.7503 

10 

—  4-1057 

+  4.9831 

+  1.1038 

—274981 

—  36.8202 

ii 

—  1.9060 

+  2.2816 

+  04940 

—29.9252 

—39.1222 

70  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

TABLE   X. 
COEFFICIENTS  OF  cos  iSL  AND  SIN  i8L  FOR  A,  B,  C,  AND  D  IN  EACH  OF 

12    SYSTEMS  (L  =  V  —£"')• 


System. 

3oAc 

3°^. 

30^e 

30^. 

3oC. 

30  c. 

3oA 

3oA 

o 

+6.2049 

+0.6020 

—4.9971 

-0.5329 

—1.2077 

—0.0692 

+0.5720 

-54864 

I 

+5-1524 

+1.6797 

—4.2291 

—14580 

—0.9233 

—0.2217 

+1-5716 

—4-6133 

2 

+4.0908 

+2.0174 

—3-4091 

—1.7277 

-0.6816 

—0.2896 

+1-8673 

—3-7002 

3 

+3-5721 

+1.8476 

—2.9712 

—1-5447 

—0.6009 

—0.3028 

+1-6835 

—3.2281 

4 

+3-5031 

+1.3168 

-2.8674 

—1.0770 

-0.6358 

—0.2398 

+1.1840 

—3-1341 

5 

+3-5273 

+0.4896 

-2.8577 

—04074 

—0.6697 

—0.0821 

.  +0.4448 

—3-1352 

6 

+3-3871 

—04122 

—2.7680 

+0.3160 

—0.6191 

+0.0962 

—0.3565 

—3-0267 

7 

+3-1874 

—1.1332 

—2.6537 

+0.9305 

-0.5337 

+0.2028 

—1.0208 

-2.8813 

8 

+3-2562 

—1.6267 

—2.7313 

+  1.3868 

—0.5250 

+0.2401 

-1.2991 

—2.9577 

9 

+3-8775 

—1.9181 

—3-2097 

+1.6630 

-0.6677 

+0.2552 

-1.7883 

—34938 

10 

+5.0768 

—1-7932 

—4.1114 

+I-550I 

—0.9655 

+0.2431 

—1.6730 

-4-5098 

ii 

+6.2050 

—0.8671 

—4.9641 

+0.7360 

—1.2409 

+0.1311 

—0.8012 

-54650 

a. 

+4-2534 

+0.0169 

-3-4808 

—0.0138 

—0.7726 

—0.0031 

+0.0154 

—3.8026 

at 

+1-3404 

+04237 

—1.0631 

—0.3618 

—0.2773 

—  0.0620 

+0.3915 

-1.1728 

£. 

—0.2170 

+1.9499 

+0.1657 

-1.6577 

+0.0512 

—0.2923 

+1.7968 

+0.1855 

at 

+0.5358 

+0.0646 

—0.3961 

—0.0834 

—0.1397 

+0.0188 

+0.0796 

—04480 

0» 

—0.3789 

+0.2585 

+0.2565 

—0.2410 

+0.1224 

—0.0174 

+0.2547 

+0.3019 

ai 

+0.0682 

+0.0800 

—0.0512 

—0.0602 

—0.0170 

—  0.0107 

+0.0698 

—0.0569 

0. 

—0.0679 

+0.0673 

+0.0488 

—0.0540 

+O.OIO2 

—0.0133 

+0.0611 

+0.0554 

The  coefficients  Ac,  A,,  etc.,  have  a  separate  value  for  each  ol  the  12  systems. 
These  special  values  are  developed  in  a  periodic  series  proceeding  according  to  the 
sines  and  cosines  of  multiples  of  g"',  in  the  form  (a)  §36  with  results  shown  in  the 
last  seven  lines  above.  The  final  development  is  then  shown  below  in  the  form  (b}. 


TABLE  XI. 

COMPUTATION  OF  A-  AND  K-COEFFICIENTS  FOR  THE  HANSENIAN  INEQUALITY  OF 

LONG  PERIOD. 


Arg. 

30^ 

3oAt 

30^c 

302?. 

30  cc 

30  c. 

y>*>. 

3oZ>. 

iSv-iSg7 
i8v—  17^ 
i8v—  16^ 
iSv-is?' 

+4-2534 
—0.3048 
+0.1387 
+0.0005 

+0.0169 
+0.1034 
—0.1572 
+0.0061 

-3.4808 

+0.2973 
—0.0775 
+0.0014 

—0.0138 
—  0.0981 
+0.0865 
—0.0057 

—0.7726 
+0.0075 
—  0.06  1  1 
—  0.0019 

—0.0031 
—0.0054 
+0.0706 
—0.0003 

+0.0154 
+0.1030 

—  o.nu 
+0.0072 

-3.8026 
+0.3120 
—0.0966 

+O.002I 

Arg. 

30*; 

30*; 

\<?MKe 

i<?MKt 

io*MCc 

io3MCt 

io3AfDc 

ioWZ>. 

i8v—  18/ 
i8v—  17^ 
i8v—  16^ 
i8v-is«' 

+3.8671 
—0.3010 

+0.1081 
—0.0004 

+0.0104 
+0.1008 

—  0.1218 
+0.0059 

+0.5469 
—  0.0426 
+0.01528 
—  0.00006 

+0.0015 
+0.0142 
—0.01723 
+0.00083 

—0.1093 

+0.00  1  1 

—  0.00864 
—0.00027 

—0.0004 
—  0.0008 
+0.00998 
—0.00004 

+O.OO22 
+O.OI46 
—O.OI57I 
+O.OOI02 

-0^5376 
+0.0441 
—0.01366 
+0.00030 

COEFFICIENTS   FOR  DIRECT  ACTION. 


§37.  Coefficients  E  and  F  for  Venus.  Some  preliminary  computations  ren- 
der it  doubtful  whether  the  planetary  coefficients  E  and  F  would  lead  to  sensible 
inequalities  in  any  case.  But,  in  order  to  leave  no  doubt,  they  are  computed  for 
six  ot  the  twelve  systems  and  thirty  alternate  values  of  the  index  for  Venus.  The 
separate  numerical  results  are  shown  in  Table  XII.  The  general  development 
will  be,  so  far  as  it  seemed  useful  to  use  it,  found  in  Part  IV. 

TABLE   XII. 
SPECIAL  VALUES  OF  E  AND  F  FOR  THE  ACTION  OF  VENUS  ON  THE  MOON. 


Coefficient  E. 

Coefficient  F. 

i 

System 
O 

System 

2 

System 
4 

System 
6 

System 

8 

System 
10 

l 

System 

0 

System 

2 

System 
4 

System 
6 

System 
8 

System 
IO 

O 

+3.528 

+6.997 

+3.567 

—2.569 

—6.300 

-4.684 

o 

+0.071 

+0.617 

+0.231 

+0.039 

+0.532 

+0.345 

I 

+2.2  16 

+3-683 

+  1.277 

-1.664 

—2.541 

—1.248 

i 

-I.I54 

—1.613 

—0.556 

+0.836 

+1484 

+0.750 

2 

+0.643 

+0.872 

+0.160 

—0.545 

—0.620 

—0.150 

2 

—0.571 

+0.701 

—0.126 

+0.459 

+0.574 

+0.144 

3 

+0.218 

+0.233 

—0.006 

—0.198 

—0.179 

o.ooo 

3 

—0.229 

—0.226 

—0.005 

+0.198 

+0.192 

o.ooo 

4 

+0.094 

+0.078 

—0.024 

—0.089 

—0.064 

+0.019 

4 

—  O.IOO 

-0.077 

+0.023 

+0.090 

+0x169 

—  O.02I 

5 

+0.049 

+0.030 

—  O.022 

—  0.046 

—0.026 

+0.018 

5 

—0.048 

—0.028 

+O.02O 

+0.045 

+0.026 

—0.019 

6 

+0.028 

+O.OI2 

—O.OlS 

—  0.027 

—  0.024 

+0.015 

6 

—0.025 

—  O.OIO 

+0.015 

+0.024 

+O.OIO 

—0.014 

7 

+0.017 

+0.004 

—  O.OI4 

—0.017 

—  0.004 

+0.013 

7 

—  0.014 

+0.003 

+O.OI  I 

+0.013 

+0.003 

—  O.OIO 

8 

+O.OII 

0.000 

—0.012 

—  O.OII 

o.ooo 

+O.OII 

8 

—0.007 

0.000 

+0.007 

+0.007 

o.ooo 

—0.007 

9 

+0.007 

—0.002 

—0.009 

—0.007 

+0.002 

+0.009 

9 

—0.004 

+O.OOI 

+0.005 

+0.004 

—0.001 

—0.005 

10 

+0.004 

—0.003 

—0.008 

—0.004 

+0.003 

+0.008 

10 

—0.002 

+O.002 

+0.003 

+O.OO2 

—O.OOI 

—0.004 

ii 

+0.003 

—  O.OO4 

—O.OO7 

—0.003 

+0.004 

+0.006 

ii 

—O.OOI 

+O.OOI 

+O.OO2 

+O.OOI 

—O.OOI 

—  O.OO2 

12 

+O.OOI 

—  O.OO4 

—0.005 

—  O.OOI 

+0.004 

+0.005 

12 

0.000 

+0.001 

+0.001 

0.000 

—0.001 

—0.002 

13 

o.ooo 

—0.005 

—0.005 

0.000 

+0.004 

+0.004 

13 

o.ooo 

+O.OOI 

+O.OOI 

o.ooo 

—O.OOI 

—  O.OOI 

14 

—0.00  1 

—0.005 

—0.004 

+0.001 

+0.005 

+0.004 

14 

0.000 

0.000 

0.000 

o.ooo 

—O.OOI 

o.ooo 

IS 

—  O.OO2 

—0.005 

—O.OO3 

+O.OO2 

+0.005 

+0.003 

IS 

0.000 

o.ooo 

0.000 

0.000 

0.000 

0.000 

16 

—0.003 

—0.005 

—  O.OO2 

+0.003 

+0.005 

+O.OO2 

16 

o.ooo 

—O.OOI 

o.ooo 

o.ooo 

o.ooo 

o.ooo 

17 

—  0.004 

—0.005 

—  0.001 

+0.004 

+0.005 

+0.001 

17 

—  O.OOI 

—O.OOI 

o.ooo 

+O.OOI 

+O.OOI 

o.ooo 

18 

—0.005 

—0.005 

O.OOO 

+0.005 

+0.005 

0.000 

18 

—O.OOI 

—O.OOI 

0.000 

+O.OOI 

+0.001 

0.000 

19 

—0.006 

—0.005 

+O.OOI 

+0.006 

+0.005 

—  O.OO2 

19 

—  O.OO2 

—  O.OO2 

o.ooo 

+O.OO2 

+O.O02 

—O.OOI 

20 

—0.008 

—0.004 

+0.003 

+0.007 

+0.005 

—0.003 

20 

—0.004 

—  O.OO2 

+O.OOI 

+0.003 

+O.OO2 

—  O.OO2 

21 

—0.0  10 

—0.004 

+0.005 

+0.009 

+0.004 

—0.006 

21 

—0.006 

—  O.002 

+0.003 

+0.005 

+0.002 

—0.003 

22 

—0.012 

—0.003 

+0.009 

+O.OI2 

+0.002 

—  O.OIO 

22 

—0.008 

—O.OOI 

+0.006 

+0.008 

+O.OO2 

—0.007 

23 

—  0.016 

o.ooo 

+0.014 

+0.015 

o.ooo 

—0.017 

23 

—  O.OI2 

o.ooo 

+0.01  1 

+O.OI2 

o.ooo 

—  O.OI2 

24 

—  O.O2I 

+0.005 

+0.024 

+O.020 

—0.007 

—  0.029 

24 

—0.019 

+0.005 

+0.022 

+0.017 

—0.006 

—0.025 

25 

—0.028 

+0.018 

+0.043 

+0.027 

—0.023 

—0.053 

25 

—0.028 

+0.018 

+0.043 

+0.026 

—  O.02I 

—0.050 

26 

—0.039 

+0.050 

+0.084 

+0.037 

—0.065 

—0.108 

26 

—0.042 

+0.055 

+0.090 

+0.038 

—  0.064 

—0.109 

27 

—0.050 

+0.152 

+0.194 

+0.045 

—0.208 

—0.264 

27 

—0.053 

+0.167 

+0.205 

+0.044 

—0.198 

—  0.260 

28 

o.ooo 

+0.569 

+0.561 

—0.017 

—0.799 

—0.819 

28 

0.000 

+0-543 

+0.507 

—0.014 

—0.623 

—0.672 

29 

+0.730 

+2.564 

+1.885 

-0.681 

—3.325 

—2.919 

29 

+0.406 

+  1-557 

+1.061 

—0.325 

—1.402 

—1.331 

B.    ACTION  OF  MARS. 

§38.  For  Mars  the  coefficients  A,  B,  C,  and  D  were  developed  much  in  the  same 
way  as  for  Venus.  But,  owing  to  the  supposed  absence  of  terms  having  a  high  mul- 
tiple of  the  mean  longitude  of  Mars,  it  was  considered  sufficient  to  divide  the  mean 
orbit  of  Mars  into  24  parts  for  the  special  computations  of  the  yl-coefficients.  The 
adopted  number  of  systems  was  12,  as  in  the  case  of  Venus. 

The  following  statements,  with  the  diagram,  will  make  clear  the  method  of  carry- 
ing out  the  computation.  In  system  o  the  Earth  remains  at  rest  at  its  perihelion 


72 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


and  is,  therefore,  in  longitude  approximately  TTO'  =  99°.5.  Mars  starting  from  this 
same  mean  longitude,  TTO',  takes  the  twenty-four  consecutive  mean  longitudes  TTO', 
ITO'  -f- 15°,  TTO'  -j-  30°,  etc.,  to  if1'  -f-  345°.  These  twenty-four  positions  are  designated 
by  the  twenty-four  indices  o,  i,  2,  3,  ...  23. 

In  system  i  the  Earth  is  in  mean  anomaly  30°.  Then,  as  before,  Mars  takes  the 
successive  mean  longitudes  TTO'  +  30°,  TTO' -\-  45° ,  .  .  .  up  to  TTO'  -f- 15°. 

The  same  plan  is  carried  through;  the  constant  mean  anomaly  of  the  Earth  in 
the  tth  system  being  i  X  30°,  while  Mars,  starting  with  the  same  mean  longitude, 
goes  through  its  twenty-four  consecutive  mean  positions,  the  indices  which  express 
the  mean  longitude  of  Mars  always  starting  with  the  value  o  when  Mars  is  in  mean 
conjunction  with  the  Earth. 

As  in  the  case  of  Venus,  the  elements  were  taken  with  their  values  for  1800,  in 
order  to  correspond  to  the  mean  of  the  period  during  which  the  longitude  of  the 
Moon  has  been  observed.  The  numbers  and  data  for  computing  the  longitude  of 
Mars  are,  then,  as  follows: 

TTO' ;  long,  of  ®'s  perihelion  for  1800;  ....     99°  30'     7". 6 
7T4;       "      "  Mars'      "  "       ";....  332    22    42   .9 

7r0'-7r4;  initial  mean  anom.  of  Mars  for  1800;  127      7    24  .7 
Initial  mean  anomaly  of  Mars  in  system  j 


x/ 


M0 


For  system  j  and  index  i 


Equinox 


7'24".7  +  3o0  x/+  15°  x  i 

From  the  numbers  found  in  Tables 
of  MarS)  page  397,  it  is  found  that  to 
this  initial  mean  anomaly  corresponds 

Fund.  Arg.  N=  243^.0948 

and  that  the  increment  of  JVfor  15°  of 
mean  anomaly  is 


Arrangement  of   Coordinate  Axes  in  Systems  o,  i,  etc., 

for  Mars.  have : 


For  the  numbers  arising  from  the 
inclination    of   the    orbit    of   Mars   we 


Long,  of  node,  1800;  .  6  =  48°  o'  52". 5 
Node  from  e's  perihelion  ;  308°  30' 44". 9 
Inclination,  1800;  ....  /=i°5i'  3". 6 


COEFFICIENTS   FOR  DIRECT  ACTION. 


73 


The  results  of  the  main  steps  in  the  computation  of  the  coordinates  of  Mars  are 
shown  in  the  following  table.  The  first  column  corresponds  to  the  indices  of 
system  o.  In  they'th  system  they  are  diminished  by  2;. 

The  second  column  shows  the  value  of  N  actually  used  in  entering  the  tables. 
The  discrepancy  of  two  units  in  the  fourth  place  results  from  using  two  computa- 
tions of  N.  Columny  gives  the  mean  anomaly  as  taken  from  the  tables,  reduced 
by  the  secular  variation  to  1800.  Column  u  is  formed  by  adding  to^  the  distance 
from  the  node  to  the  perihelion  of  Mars  and  applying  the  reduction  to  the  ecliptic. 
This  reduction  was  applied  in  order  to  use  for  x  and  y  simple  formulae  for  the 
ecliptic  longitude.  Actually,  through  a  misapprehension,  the  rectangular  coordi- 
nates were  computed  on  the  supposition  that  u  was  counted  along  the  orbit,  as  in 
the  case  of  Venus.  There  is  therefore  an  error  in  the  last  figures  of  the  coordinates, 
the  amount  of  which  can  readily  be  determined,  but  which  has  been  deemed  too 
small  to  need  correction  for  the  present  problem. 


TABLE   XIII. 
COMPUTATION  OF  HELIOCENTRIC  COORDINATES  OF  MARS. 


i 

N 

/ 

U 

log.  r 

X 

y 

z 

o 

243.0946 

0     i      it 

135   3  17 

59  24  20 

0.208  749 

—  1.601  16 

—  O.222  32 

+0.04497 

i 

271.7194 

148   7   I 

72  28  19 

0.214918 

—1.53086 

—0.58691 

+0.050  53 

2 

300.3442 

160  52  ii 

85  13  52 

0.219  143 

—1.37661 

-0.919  51 

+0.053  32 

3 

328.0600 

173  26   7 

07  48  12 

0.221  310 

—1.14904 

—1.20324 

+0.053  27 

4    357-5939 

185  56  13 

no  18  38 

O.22I  364 

—  0.86  1  21 

-1.42386 

+0.05043 

5 

386.2187 

198  29  47 

122   52   26 

0.219  303 

—0.528  30 

-1.56981 

+0.04495 

6 

414.8435 

211   14   12 

135  36  56 

0.2I5I79      —O.I6745 

—1.63228 

+0.03708 

7 

4434683 

224  16  54 

148  39  31 

O.209  102      +0.202  38 

—  1-60553 

+0.027  19 

8 

472.0932 

237  45  02 

162   7  24 

0.201  264      +0.560  38 

—148737 

+0.015  76 

9 

500.7180 

251  45  55 

176   7  52 

0.191962      +0.88448 

—  1-27995 

+0.00339 

10 

529.3428 

266  25  53 

190  47  23 

O.lSl  634 

+1-151  73 

—0.090  76 

—  0.009  19 

ii 

557-9676 

281  49  50 

206  10  58 

0.170884 

+1.33964 

-0.633  72 

—  O.O2I  13 

12 

586.5024 

298   o  12 

222   21    9 

0.160502 

+142836 

—0.230  06 

—0.031  49 

13 

615.2172 

314  55  23 

239  16  26 

O.I5I4I9 

+  1.40360 

+0.191  54 

-0.03935 

14 

643.8421 

332  28  46 

256   50   12 

0.144609 

+1.26023 

+0.50686 

—0.043  88 

15 

672.4669 

350  28  12 

274  50  II 

O.I40  9IO 

+1-00534 

+0.94909 

—0.044  52 

16 

14.0061 

8  37  13 

292  59  42 

O.I408I7 

+0.65923 

+  1.21507 

—  0.041  12 

17 

42.7209 

26  37  36 

311  oo  19 

0.144343 

+0.253  23 

+  1.37065 

—0.033  99 

18 

71-3457 

44  12  35 

328  35  12 

O.I5IOI4 

—0.17528 

+  1.40475 

—0.023  84 

19 

09.9705 

61   9  53 

345  32   8 

O.I6OOO6 

-0.58917 

+1.31988 

—  0.011  66 

20 

128.5954 

77  22  34 

I  44  20 

0.170346 

—0.956  46 

+1.12979 

+0.001  45 

21 

157  2202 

92  48  50 

17  10  10 

O.lSl  O98 

—1.25306 

+0.85563 

+0.01447 

22 

185.8450 

107  30  54 

31  Si  56 

O.I9I  465 

—1.46347 

+0.522  13 

+0.026  50 

23 

214.4608 

121  33  36     45  54  32 

0.200829      —1.57994 

+0.154  73 

+0.036  84 

74 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE    XIV. 

G-COORDINATES    OF    MARS    REDUCED    TO    THE    DIFFERENT    SYSTEMS. 


System  o. 

System  i. 

System  2. 

System  3. 

jr 

r 

X 

r 

X 

r 

X 

r 

o 
I 

2 

3 

4 

—0.617  95 
-0.54765 
—0.393  40 
—0.165  83 

+O.I220O 

—0.22232 

—0.586  91 
—0.919  Si 
—1.203  24 
—142386 

—0.666  54 
—0.61133 
—047237 
—0.25703 
+0.024  24 

—0.091  17 
—0450  67 
—0.785  65 
—  1.07849 
—1-31303 

—0.672  31 
—0.632  25 
—0.505  93 
—0.297  85 
—  0.016  52 

+0.063  05 
—0.298  23 
—0.641  98 
—0.948  87 
—1.19984 

—0.632  56 
—0.605  8  1 
—0487  65 
—0.280  23 
+0.00896 

+O.2OI  03 
—O.l688o 
—O.526  8O 

—0.850  go 
—1.11815 

5 
6 

8 
9 

+0.45491 
+0.8I5  76 
+  I-I85S9 

+  1-54359 
+  1.86769 

-1.56981 
—1.63228 
-1.60553 
-1.48737 
-1-27995 

+0.35789 

+0.727  02 
+  I.III  41 
+  1.48744 
+  1.82870 

—147477 

—I-35I  44 
-1.53386 
—  1.41703 
—  1.201  79 

+0.325  iC 
+0.709  23 
+1.11239 
+  1-50633 
+  1-85907 

-1.37681 
—146366 
—144788 
—1.32288 
—1.09063 

+0.366  oo 
+0.76966 
+1.191  26 
+  1-59658 
+1.94881 

—1.30606 
—1.39478 
—1.37002 
—1.22665 
—0.971  76 

10 
ii 

12 

13 

14 

+2.134  94 
+2.322  85 
+2.41157 
+2.38681 
+2.243  44 

—0.99076 
—0.633  72 
—  0.230  06 
+0.191  54 
+0.59686 

+2.107  36 

+2.29671 

+2.375  22 

+2.330  58 
+2.16383 

-0.89657 
—0.51907 
—0.096  37 
+0.336  12 
+0.739  52 

+2.13839 
+2.31600 
+2.373  28 

+2.305  02 
+2.I203O 

—0.763  82 
—0.366  96 

+0.065  77 
+0495  16 
+0.883  32 

+2.214  79 
+2.370  37 
+2404  47 
+2.31960 
+2.129  51 

—0.625  65 
—0.219  65 
+0.20886 
+0.622  75 
+0.990  04 

IS 
16 
17 
18 
19 

+  1.98855 
+1.64244 
+  1.23644 
+0.80793 
+0.39404 

+0.04900 
+  1.21507 
+1-37065 
+  1-40475 
+1-31988 

+1.89002 

+1.53596 
+1-13509 
+0.721  96 
+0.32801 

+  1.07726 
+  1.32104 
+  1.45449 
+  I-4735I 
+  1-38438 

+  1.83987 

+I-49I  59 
+  1.10586 
+0.71184 
+0-335  42 

+I-I9933 
+  1.42236 
+  1-542  IS 
+i.5576i 
+  1.47478 

+  1-85535 
+  1.52185 
+I-I5445 
+0.777  40 
+0.41281 

+1.28664 
+1-49705 
+1.61352 
+1.63474 
+1.56444 

20 
21 
22 
23 

+0.026  75 
—0.26985 
—  0.480  26 
—0.596  73 

+1.12979 
+0.85563 
+0.522  13 
+0.154  73 

—  0.020  95 

—0.305  52 
—0.51242 
—0.63383 

+  1.20076 
+0.040  82 
+0.624  89 
+0.274  oo 

—o.oo  i  73 
—0.282  32 
—0.493  23 
—0.625  17 

+  1.30464 
+  1.06147 
+0.761  58 
+0.422  63 

+0.08021 
—0.203  52 
—0424  14 
—0.570  09 

+1.410  19 
+1.18262 
+0.894  79 
+0.561  88 

System  4. 

System  5. 

System  6. 

System  7. 

O 

I 

2 
3 

4 

—0.560  10 
—0.542  52 
—0.425  69 
—  0.210  45 
+0.094  77 

+0.28739 
—0.09700 
—0473  03 
—0.814  29 
—1.092  95 

—0478  34 
—0462  56 
—0.337  56 
—0.105  31 
+0.221  50 

+0.29889 
—0.104  27 

—0.498  21 

—0.850  95 
—1.13027 

-0.4II57 
—0.38681 
—0.243  44 
+0.0  1  1  45 
+0.357  56 

+0.230  06 
—0.191  54 
-0.59686 
—0.94909 
—1.21507 

—0.37S  36 
—0.330  72 
—0.16397 
+0.10984 
+0.463  90 

+0.09649 
—0.336  oo 
—0.73940 
—1.07714 
—1.32092 

S 
6 

8 
9 

+0472  27 
+0.89497 
+1.32746 
+1.73086 
+2.06860 

—1.282  30 
—1.36081 
—1.316  17 
—1.14942 
—0.875  61 

+0.618  36 
+1.05109 
+148048 
+1.86864 
+2.18465 

-1.30788 
—1.365  16 
—1.29690 

—  1.  112  l8 

-0.831  75 

+0.763  56 
+1.19207 
+  1.60596 
+  1.97325 
+2.26985 

—1.37065 
—1.40475 
-1.31988 
—1.12979 
—0.855  63 

+0.864  77 
+  1.27790 
+  1.67185 

+2.O2O  8l 
+2.305  38 

—  1-454  37 
—1-473  39 
—  1.38426 
—  1.20064 
—0.040  70 

10 
ii 

12 

13 
14 

+2.31238 
+2.445  83 
+2.46485 
+2.375  72 
+2.192  10 

—0.521  55 
—  0.12068 
+0.292  45 
+0.68640 
+1.03536 

+2.407  68 
+2.52747 
+2.54293 
+2.460  10 
+2.289  96 

—0483  47 
—0.097  74 
+0.296  28 
+0.672  70 
+  1.00985 

+2.480  26 
+2.596  73 

+2.61795 
+2.547  65 
+2.323  -10 

—0.522  13 
—0.15473 

+O.222  32 
+0.586  91 
+0.9I95I 

+2.51228 
+2.633  69 

+2.666  40 
+2.61  1  19 
+2.47223 

—0.624  77 
—0.273  88 
+0.091  29 
+0.450  79 
+0.785  77 

11 

17 
18 
19 

+1.932  16 
+1.61623 
+1-26534 
+0.900  17 

+0.54067 

+I.3I993 
+  1.52683 
+  1.64824 
+  1.68095 
+1.625  74 

+2.04679 
+  1.74690 
+1.40795 
+1.04837 
+0.687  09 

+1.29044 
+  I-50I  35 
+  1.63329 

+  1.68043 
+1.64037 

+2.16583 
+  1.87800 
+  1-54509 
+  1.18424 
+0.81441 

+  1.20324 
+  142386 
+  I.5698I 
+  1.63228 

+1-605  53 

+2.256  89 
+1.97562 
+  1.64197 
+  1.27284 
+0.88845 

+  1.07861 
+  I.3I3I5 
+  1.47489 
+  I.55I  56 
+  1.53398 

20 

21 
22 
23 

+0.205  69 
—0.087  15 
—0.321  69 
—048343 

+148678 
+1.271  44 
+0.090  17 
+0.656  52 

+0.343  34 
+0.03645 
—0.214  52 
—0.391  49 

+1-51405 
+  1-30597 
+  1.02464 
+0.682  96 

+0.456  41 
+0.13231 
—0.134  94 
—0.322  85 

+148737 
+  1-27995 
+0.990  76 
+0.633  72 

+0.51242 
+0.171  16 
—0.107  50 
—0.296  85 

+  I.4I7I5 

+  I.2OI  91 
+0.89669 
+0.519  19 

COEFFICIENTS   FOR  DIRECT  ACTION. 


75 


TABLE   XIV.— Concluded. 

G-COORDINATES    OF    MARS    REDUCED    TO    THE    DIFFERENT    SYSTEMS. 


System  8. 

System  9. 

System  10. 

System  n. 

^ 

f 

X 

r 

X 

r 

X 

r 

o 

I 

2 

a 

4 

—0.373  70 
—0.305  44 
—  0.12072 
+O.I597I 
+0.507  99 

—0.065  64 
—0495  03 
—0.883  19 
—1.19920 
—1.422  23 

—0.405  03 
—0.320  16 
—0.13007 
+0.14409 
+0.47759 

—0.20886 

—0.622  75 
—0.090  04 
—1.28664 
—1497  05 

—0.465  27 

—0.376  14 
—0.192  52 
+0.067  42 
+0.383  35 

—0.29258 

-0.686  53 
—1.035  49 
—1.32006 
—1.52696 

—0.54307 
—0460  24 
—  0.290  10 
—0.046  93 
+0.252  96 

—  029640 
—0.672  82 
—  1.00997 
—1.20056 

—1.501  47 

5 
6 

8 
9 

+0.893  72 
+  1.28774 
+  1.664  16 

+2.001  31 

+2.281  90 

—1.54202 
-1.55748 
—147465 
—I.3045I 
—1.06134 

+0.84499 
+  1.22204 
+1.58663 
+  1.91923 

+2.2O2  96 

—1.613  52 
—1.634  74 
—1.56444 
—  1.41019 
—1.18262 

+0.734  24 
+  1.09941 
+145891 
+  1.79389 
+2.08673 

-1.64837 
—1.681  08 
—1.62587 
—148691 
-1.271  57 

+0.591  91 
+0.95149 
+1.31277 
+1.65652 
+1.96341 

—1-63341 
—  1.680  55 
—  1.640  49 
-I.5I4  17 
—1.30609 

10 

it 

12 

13 
14 

+2.49281 
+2.624  75 
+2.671  89 
+2.631  83 
+2.505  51 

—0.761  45 
—0.422  50 
—  0.062  92 
+0.298  36 
+0.642  1  1 

+2.423  58 
+2.56953 

+2.632  oo 

+2.605  25 
+2.487  09 

—0.804  79 
—0.561  88 

—  O.2OI  O3 
+0.l688o 
+0.52680 

+2.321  27 
+2483  01 
+2.55968 
+2.542  10 
+2.425  27 

—0.99030 

—0.656  65 
—0.287  52 
+0.096  87 
+0472  90 

+2.214  38 

+2.391  35 
+2478  20 
+2.462  42 
+2.33742 

—1.02476 
-0.68308 
—0.29901 
+0.104  15 
+0.49809 

15 

16 

11 

19 

+2.29743 
+2.016  10 
+1.67442 
+1.29035 
+0.887  19 

+0.94900 
+I.I9997 
+1.37694 
+  1.46379 
+144801 

+2.27967 
+1.09048 
+1.63344 
+1.22978 
+0.808  18 

+0.850  oo 

+  I.II8I5 
+  I.3O6o6 
+  1.39478 
+  1.37002 

+2.21003 
+1.00481 
+  1-52731 
+  1.10461 

+0.672  12 

+0.814  16 
+1.09282 
+1.282  17 
+1.36068 
+1.31604 

+2.105  17 
+  1.77836 
+1.381  50 
+0.94877 
+0.51938 

+0.85083 
+M30  15 
+1.30776 
+  1.36504 
+1.20678 

20 
21 
22 
23 

+0.493  25 
+0.14051 
—0.13881 
—0.316  42 

+I.3230I 
+1.09076 
+0.763  95 
+0.36709 

+0.402  86 
+0.050  63 
—0.215  35 
—0.37093 

+  1.22665 
+0.971  76 
+0.625  65 
+0.21965 

+0.268  72 
—0.069  O2 
—O.3I2  8O 
—0.446  25 

+1.14929 
+0.87548 
+0.521  42 
+0.120  55 

+O.I3I  22 

—0.18479 
—0407  82 
—0.527  61 

+  1.11206 
+0.831  63 
+0483  35 
+0.097  62 

TABLE  XV. 
SPECIAL  VALUES  OF  A,  B,  C,  AND  D  FOR  MARS. 


System  o. 

System  i. 

i 

A 

B 

C 

D 

i 

A 

B 

C 

D 

0 

I 

2 

3 

4 

+  1.92041 
+0.250  33 
—0.17822 
—0.175  IS 

—O.I  1  1  SO 

—0.768  57 
+0.382  65 
+0.507  19 
+0.35960 
+0.225  06 

—1.15195 
—0.63303 
—0.32897 
—0.18443 
—0.11356 

+i.iii  16 
+0.954  82 
+0.358  95 
+0.075  13 
—0.029  05 

O 
I 

2 

3 
4 

+2.08860 

+0.705  52 

-0.08866 

—  0.204  72 

—0.14686 

—1.02497 
+0.038  62 

+0.515  53 
+0.447  51 
+0.293  Si 

—1.06403 
—0.744  19 

—  0.426  83 
—  0.242  79 
—0.146  66 

+043399 
+1.07690 
+0.56893 
+0.16480 
—0.008  13 

6 

8 
9 

—0.058  53 
—  0.021  97 
+0.002  45 

+0.01880 
+0.029  90 

+0.13450 
+0.076  72 
+0.039  44 
+0.01503 
—  o.ooi  19 

—0.076  06 
—0.054  76 
—0.041  89 
—0.033  83 
—  0.028  72 

—  0.06  1  07 
—0.065  74 
—0.060  08 
—0.050  72 
—  0.040  17 

5 
6 

8 
9 

—0.079  42 
—0.030  47 
+0.001  61 

+O.O22  O2 

+0.034  83 

+0.17465 
+0.096  73 
+0.04744 
+0.016  43 
—0.003  03 

—0.095  23 
—0.066  25 
—0.049  04 
—0.038  44 
—  0.031  80 

—0.065  51 

—0.076  39 
—0.069  91 
—0.057  60 
—0.043  79 

10 
it 

12 

13 

14 

+0.037  54 
+0.042  79 
+0.046  23 
+0.048  05 
+0.04795 

—0.01198 
—  0.01892 
—0.02281 
—0.023  80 
—0.021  35 

—0.025  57 
—0.023  87 
—0.023  43 
—0.024  25 
—0.026  60 

—0.029  29 
—0.018  19 
—0.006  65 
+0.005  80 
+0.01984 

10 
ii 

12 
13 

14 

+0.042  73 
+0.04731 
+0.049  46 
+0.049  46 
+0.04697 

—0.014  99 

—  O.O2I  8  1 

—  0.024  68 
—0.023  96 
—0.019  12 

—0.027  72 
—0.025  51 
—  0.024  77 
—0.025  49 
—0.027  83 

—0.029  08 
—0.01646 
—0.003  01 
+0.0108  1 
+0.025  58 

15 
16 
17 
18 
19 

+0.044  92 
+0.036  65 
+0.01830 
—0.019  93 
—0.006  22 

—0.013  81 
+0.002  37 
+0.034  61 
+0.098  16 
+0.223  72 

—0.031  10 
—0.03901 
—0.052  92 
—0.078  25 
—0.127  50 

+0.036  30 
+0.056  01 
+0.079  oo 
-fo.ioi  49 
+0.10486 

11 
17 
18 

19 

+0.040  91 
+0.029  03 
+0.007  '9 
—0.031  63 
—0.097  28 

—0.008  56 

+O.OI  I  O4 

+0.045  88 
+0.10709 
+0.212  09 

—0.032  35 
—  0.040  06 
—0.053  06 
—0.075  45 
—0.11569 

+0.041  77 
+0.05944 

+O.077  22 

+0.08944 

+0.077  88 

20 
21 
22 
23 

—0.230  56 
—0.33621 
+0-347  58 
+2.549  oo 

+0461  52 
+0.797  25 
+0.580  43 
—1.148  74 

—0.23095 
—  0.461  05 
—0.927  87 
—1.400  16 

+0.016  40 
—0.30696 
—1.39090 
—1.02700 

20 
21 
22 
23 

—0.192  14 
—0.245  44 
+0.12735 
+  1.52082 

+0.384  18 
+0.58762 
+0495  62 
—0.53246 

—0.192  04 

—  O.342  21 

—  0.622  oo 

—0.98831 

—  o.oio  06 
—0.302  42 
—0.922  oo 
—1.091  63 

76 


ACTION   OF  THE   PLANETS  ON   THE   MOON. 


TABLE   XV .—Continued. 
SPECIAL  VALUES  OF  A,  B,  C,  AND  D  FOR  MARS. 


System  2. 

System  3. 

i 

^4 

B 

C 

D 

i 

A 

B 

C 

D 

0 

i 

2 

3 

4 

+2.IOI  41 
+  I40O  17 
+0.08945 

—0.247  49 
—0.192  78 

—  1.045  67 
—0.44244 
+0.515  34 
+0.585  27 
+0.385  57 

—1.055  73 
—0.957  75 
—  0.604  76 
—0.337  75 
—0.192  79 

—0.297  76 
+1.11788 

+0.885  66 
+0.289  97 
+0.007  96 

o 
i 

2 

3 

4 

+1-94747 
+2.37680 
+0.345  55 
-0.32752 
—0.238  32 

-0.823  38 
—1.04803 
+0.55368 
+0.791  16 
+047668 

—1.12408 
—1.32882 
-0.899  15 

—0.463  59 
—0.238  32 

—0.979  52 
+1.03457 
+1.34603 
+0.41324 

—0.005  73 

6 

8 
9 

—0.09908 
—0.033  27 
+0.00621 
+0.028  67 
+0.04097 

+0.21683 
+0.1  10  73 
+0.048  52 
+0.012  65 
—0.007  72 

—O.H774 
-0.077  45 
-0.054  73 
—0.041  33 
—0.033  24 

—0.079  02 
—  0.091  19 
—0.07934 
—0.06  1  50 
—0.043  55 

6 

I 
9 

—0.104  37 
—0.024  71 
+0.01621 
+0.036  14 
+0.045  21 

+0.237  83 
+0.10704 
+0.039  39 
+0.00461 
—0.013  oo 

—0.13343 
—  0.082  30 
—0.055  58 
—0.04075 

—0.032  22 

—0.10406 

—0.10454 
—0.082  65 

—0.059  12 

—0.038  63 

10 
ii 

12 
13 

14 

+O.O47  22 

+0.049  75 
+0.04971 
+0.04748 
+0.042  80 

—0.01880 
—0.023  94 
—0.024  83 
—  0.022  06 
—0.015  30 

—0.02843 
—0.025  8  1 
—0.024  87 
—0.025  41 
—0.027  49 

—0.027  03 
—0.01198 

+O.O02  O7 

+0.015  66 
+0.02929 

10 

ii 

12 
13 
14 

+0.048  58 
+0.048  76 
+0.046  87 
+0.043  26 
+0.037  75 

—  O.O2I  27 
—O.024  O7 

—0.023  1  8 

—  O.OI9  21 
—  O.OI2  O2 

—0.027  30 

—  0.024  68 
—0.023  70 
—0.024  06 
—0.025  74 

—  0.021  44 
—  0.006  8  1 
+0.006  13 
+0.01807 
+0.029  52 

IS 
16 
17 
18 
19 

+0.034  78 
+0.021  75 

+O.OOOO2 

—0.031  96 
—0.082  07 

—0.003  32 
+0.016  33 
+0.04785 
+0.09826 
+0.178  16 

—0.031  46 
—0.038  07 
—0.048  76 
—0.066  29 
—0.09609 

+0.043  18 
+0.05704 
+0.06929 

+0.075  22 
+0.062  41 

15 
16 

19 

+0.029  7O 
+0.017  96 
+0.00066 
—0.025  10 
—0.063  25 

—0.000  75 
+0.016  27 
+0.041  06 
+0.081  14 
+0.141  60 

—0.028  96 
—0.03424 
—0.042  62 
—0.056  05 
+0.078  35 

+0.040  68 
+0.05137 
+0.060  55 
+0.065  29 
+0.058  10 

20 
21 
22 
23 

—0.149  84 

—  O.2OI  17 

—0.051  76 
+0.807  50 

+0.299  14 
+045032 
+0491  04 
—0.049  04 

—0.14931 
—0.249  17 
—043933 
—0.75848 

—O.OOO6O 
—0.18647 
—0.605  51 
-1.06638 

20 
21 
22 
23 

—0.11689 
-0.175  77 
—0.15423 

+0.334  45 

+0.23443 
+0.366  06 
+0493  56 
+0.306  39 

—O.II753 
—0.19120 
—0.33931 
—0.640  8  1 

+O.O2O  OS 

—0.096  25 

—0.306  04 

—0.96723 

System  4. 

System  5. 

o 

I 
2 
3 
4 

+  1.832  10 
+3-797  02 
+0.44209 
—0.454  66 
—  0.246  47 

—0.50077 
—  1.806  10 
+0.849  47 
+  1.01321 
+049794 

—I-33I  96 
—1.99000 
—  1.29202 
—0.558  66 
-0.251  49 

—  1.62480 
+  1.03488 
+  1.92787 
+0.406  52 
—0.065  O4 

O 
I 

2 

3 

4 

+2.148  15 
+5-764  33 
—0.089  37 
—0.503  27 
—0.19356 

—0.293  03 
-2.665  97 
+  1.600  16 
+1.02700 
+0.410  28 

-1.855  19 
-3-098  75 
—  1.51072 
-0.523  75 
—0.21673 

—2.502  41 

+2.0O2  IO 
+2.II620 
+0.192  32 
—0.12306 

5 
6 

8 
9 

—0.083  76 
—0.00731 
+0.026  10 
+0.040  16 
+0.045  38 

+0.213  91 
+0.084  19 
+0.024  79 
—0.003  05 
—0.01600 

—0.130  15 
—0.076  89 
—0.050  89 
—0.037  10 
—0.02938 

—O.I2683 
—O.IO605 
—O.076  42 

—0.051  33 
—0.031  65 

6 

8 
9 

—0.040  74 
+0.007  54 
+0.030  46 
+0.039  37 
+0.042  28 

+0.15937 
+0.057  46 
+0.013  21 
—0.006  08 
—  0.016  18 

—  0.10963 
—0.065  oi 
—  0.043  66 
—0.032  40 
—  0.020  09 

—0.12733 
—0.00438 
—0.064  96 
—  O.O42  72 
—0.026  O3 

10 
ii 

12 
13 

14 

+0.046  40 
+0.045  24 
+0.042  68 
+0.038  98 
+0.033  97 

—0.021  39 
—0.022  53 
—0.02089 
—  0.016  95 
—  0.010  59 

—0.025  oi 

—  O.022  70 
—  O.02I  80 
—  O.O22  O4 
—0.023  38 

—  0.016  ii 
—0.003  35 
+0.007  65 
+0.017  63 
+0.027  09 

10 

ii 

12 

13 
14 

+0.042  40 
+0.041  09 
+0.038  92 
+0.035  97 
+0.032  ii 

—0.019  89 
—0.02051 
—0.019  06 
—0.015  89 
—  o.oio  87 

—  O.022  51 
—  O.O2O  60 

—0.01985 

—  O.020  07 
—  O.O2I  23 

—O.OI3  03 

—  0.002  39 
+0.006  85 
+0.015  33 
+0.023  53 

IS 
16 

H 

19 

+0.027  18 
+0.017  72 
+0.004  14 
—0.015  94 
—0.046  43 

—  o.ooi  19 
+0.012  55 
+0.032  90 
+0.063  85 
+0.11241 

—0.025  99 
—0.030  27 
—0.037  05 
—0.04791 
—0.065  99 

+0.03551 
+0.045  37 
+0.053  75 
+0.05991 
+0.059  39 

15 
16 
17 
18 
19 

+0.026  95 
+0.01975 
+0.009  24 
—0.006  86 
—0.032  71 

—0.003  45 
+0.007  46 
+0.023  92 
+0.049  63 
+0.091  80 

—0.023  48 
—O.0272I 

—0.033  15 
—0.042  78 
—0.059  09 

+0.031  81 
+0.040  39 
+0.04924 
+0.057  70 
+0.063  25 

20 

21 
22 
23 

—  0.092  87 
—0.15848 
-0-357  45 
+0.032  96 

+0.19096 
+0.31861 
+  1.20062 
+0.579  17 

-0.09808 
—  0.160  13 
—0.843  21 
—0.612  03 

+0.040  04 
—0.032  86 
—1.00468 
—0.87868 

20 
21 
22 
23 

—0.075  96 
—0.14903 
-0.253  88 
—0.17633 

+0.16481 
+0.208  19 
+0.544  16 
+0.859  Si 

-0.08886 
—0.149  19 
—0.290  26 

—0.683  21 

+0.057  56 
+0.012  49 

—0.17474 

—0.884  34 

COEFFICIENTS   FOR  DIRECT  ACTION. 


77 


TABLE   XV '.—  Continued. 
SPECIAL  VALUES  OF  A,  B,  C,  AND  D  FOR  MARS. 


System  6. 

System  7. 

i 

^ 

B 

C 

D 

1 

A 

B 

C 

D 

o 

+4-029  84 

-0.912  73 

-3.11674 

—4.018  27 

o 

+9.99063 

—4.58621 

—5404  24 

—4.012  20 

i 

+5-687  73 

-1.695  28 

—3.992  42 

+4.843  56 

i 

+145390 

+1.601  76 

-3.055  56 

+4.666  10 

2 

—0.709  S3 

+  1.92811 

—  1.2187.) 

+1.29050 

2 

—0.656  79 

+1.41386 

-0.757  IS 

+0.48293 

3 

-0.38833 

+0.774  20 

—0.385  94 

—0.014  03 

3 

—0.254  14 

+0.515  57 

—0.261  46 

—0.07932 

4 

—0.12466 

+0.287  92 

—0.163  25 

—0.13292 

4 

—0.081  45 

+0.202  75 

—  O.I2I  32 

—0.11385 

5 

—0.024  99 

+O.III  12 

—O.O86  12 

—0.10995 

5 

—0.014  79 

+0.083  51 

—0.068  73 

—0.09063 

6 

+0.013  63 

+0.039  63 

—  0.053  26 

—0.07885 

6 

+0.012  94 

+0.032  oo 

—0.04493 

—0.066  72 

7 

+0.029  34 

+0.007  77 

—0.037  10 

—0.054  60 

7 

+0.025  42 

+0.007  18   !  —0.032  59 

—0.048  03 

8 

+0.035  71 

—0.007  35 

—0.028  35 

—0.03668 

8 

+0.031  23 

—0.00558   .  —0.02565 

—0.033  80 

9 

+0.037  98 

—0.014  63 

—0.023  35 

—O.023  12 

9 

+0.033  91 

—  0.012  35      —  0.021  58 

—  O.O22  65 

10 

+0.038  32 

—0.017  86 

—  O.O2O  46 

—0.012  38 

10 

+0.035  04 

—0.015  85      —0.019  18 

—0.013  49 

ii 

+0.037  65 

—  0.018  73 

—  O.OI892 

—0.003  37 

ii 

+0.035  30 

—0.01736   '  —0.01792 

—0.005  54 

12 

+0.03633 

—0.017  98 

—O.OI83S 

+0.004  65 

12 

+0.035  oo 

—  0.01748 

—0.017  52 

+0.001  80 

13 

+0.034  45 

—0.015  82 

—O.OI862 

+0.012  23 

13 

+0.034  24 

—  0.016  35 

—0.017  88 

+0.00900 

14 

+0.031  88 

—  0.012  14 

—0.01973 

+0.019  84 

14 

+0.032  oo 

—0.013  83 

—0.019  06 

+0.016  52 

IS 

+0.02828 

—  0.00641 

—  O.O2I  87 

+0.027  88 

IS 

+0.030  68 

—0.00942 

—  O.O2I  27 

+0.024  84 

16 

+0.023  oo 

+0.002  40 

—O.O25  42 

+0.036  73 

16 

+0.02696 

—  0.002  02 

—0.024  94 

+0.034  5i 

17 

+0.014  84 

+0.016  27 

—O.03I  II 

+0.046  70 

17 

+0.020  46 

+0.010  52 

—0.030  99 

+O.046  22 

18 

+0.001  39 

+0.039  19 

—0.040  59 

+0.05791 

18 

+0.008  52 

+0.032  70 

—0.041  23 

+0.06066 

iQ 

—  O.O22  OS 

+0.079  14 

—0.057  07 

+0.069  1  1 

19 

+0.008  53 

+0.032  71 

—0.041  24 

+0.06067 

20 

—0.065  65 

+0.154  13 

—0.08847 

+0.074  45 

20 

—0.063  61 

+0.16008 

—0.097  38 

+0.093  42 

21 

—0.154  98 

+0.315  09 

—  0.160  14 

+0.04951 

21 

-0.175  09 

+0.361  07 

—  0.186  02 

+0.07793 

22 

-0.315  17 

+0.648  46 

—0.333  30 

—0.133  72 

22 

—0432  50 

+0.882  46 

—  0450  02 

—0.15994 

23 

-0.353  82 

+1.27669 

—0.922  91 

—  1.  121  87 

23 

—0.40883 

+1.93710 

—1.528  17 

-1.99273 

System  8. 

System  9. 

0 

+  11.25307 

—5.464  71 

-5.78803 

+3.030  oo 

0 

+4.78203 

—1.30327 

-347885 

+4.27464 

i 

—  0.29587 

+  1.96385 

-1.66747 

+2.25142 

I 

—0.362  oo 

+I-33I  70 

—0.96966 

+I.I8354 

2 

—  0.44425 

+0.913  37 

—046909 

+0.189  10 

2 

+0.31773 

+0.652  46 

—0.334  77 

+0.12970 

3 

—  0.17840 

+0.366  62 

—0.18819 

—0.073  90 

3 

—0.14787 

+0.301  39 

—  0.153  52 

—0.050  95 

4 

—  0.06393 

+0.16071 

—0.096  78 

—0.091  97 

4 

—0.062  08 

+0.14789 

—0.085  80 

—0.074  57 

S 

—  0.01447 

+0.073  33      —0.058  87 

—0.076  63 

5 

—0.019  55 

+0.074  61 

—0.055  06 

-0.067  95 

6 

+  0.00879 

+0.03156      —0.04036 

—0.059  48 

6 

+0.002  94 

+0.036  19 

—0.039  12 

—0.056  34 

7 

+  0.02061 

+0.009  67 

—0.030  29 

—0.045  12 

7 

+0.015  67 

+0.014  40 

—0.030  06 

—0.045  14 

8 

+  0.02700 

—  0.002  58 

—0.024  40 

—0.033  52 

8 

+0.023  35 

+O.OOI  26 

—  0.024  62 

—0.035  28 

9 

+  0.03062 

—0.009  76 

—  O.O20  87 

—0.023  97 

9 

+0.028  30 

—0.007  oi 

—  O.O2I  28 

—0.026  63 

10 

+  0.03279 

—0.014  oo 

—0.018  79 

—  0.015  76 

10 

+0.031  67 

—0.012  37 

—O.OI9  3O 

—0.018  82 

ii 

+  0.03409 

—  0.016  39 

—0.01771 

—0.008  34 

ii 

+0.034  10 

—0.015  80 

—O.OlS  29 

—  O.OI  I  46 

12 

+  0.03485 

—  0.017  42 

—0.01743 

—  o.ooi  23 

12 

+0.035  9i 

—0.01780 

—  0.01811 

—0.004  13 

13      +  0.035  17 

—0.017  25 

—  0.01791 

+0.006  02 

13 

+0.037  22 

—0.0  1  8  50 

—0.0  1  8  72 

+0.003  63 

14 

+  0.03495 

—0.015  70 

—0.019  25 

+0.013  89 

14 

+0.03706 

—0.017  67 

—  O.O2O  28 

+0.012  31 

IS 

+  0.03390 

—  O.OI22I 

—  O.O2I  69 

+0.022  97 

IS 

+0.037  78 

—0.014  65 

—0.023  14 

+0.022  74 

16 

+  0.031  36 

—0.005  56 

—0.025  8  1 

+0.03403 

16 

+0.035  87 

—0.007  85 

—0.028  oi 

+0.035  88 

17 

+  0.02684 

+0.00688 

—  0.032  72 

+0.048  15 

17 

+0.030  24 

+0.006  33 

—0.036  58 

+0.053  56 

18 

+  0.01399 

+0.03088 

—0.044  86 

+0.066  76 

18 

+0.016  16 

+0.035  63 

—0.051  78 

+0.077  10 

19 

—  0.01233 

+0.080  37 

—0.068  02 

+0.09094 

19 

—0.018  70 

+0.101  31 

—0.082  60 

+0.10859 

20 

—  0.07503 

+0.193  18 

—0.11815 

+0.1  16  14 

20 

—0.10951 

+0.263  59 

—0.154  08 

+0.13735 

21 

—  0.23787 

+048697 

—0.249  14 

+0.094  95 

21 

—0-357  72 

+0.716  16 

—0.358  39 

+0.056  10 

22 

—  0.64095 

+  1.34280 

—0.701  91 

—0.372  76 

22           —0.781  89 

+1.91271 

—1.13082 

—  I.O52  12 

23 

+  0.77527 

+2.043  47 

-2.818  93 

-4-2S3  90 

23        +4-977  43 

-0.931  5i 

-4.045  08 

-5.38850 

8o 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


C.   ACTION  OF  JUPITER. 

§  39.  The  action  of  Jupiter  being  computed  on  the  same  general  method  as 
Venus  and  Mars,  but  being  much  simpler,  no  detailed  explanation  seems 
necessary.  Six  systems,  which  suffice  to  carry  the  coefficients  to  terms  of  the 
third  order  in  the  eccentricities,  were  deemed  enough. 

The  principal  numbers  used  or  derived  are  shown  in  the  following  tables.  The 
fundamental  data  in  the  first  table  were  derived  from  Hill's  Tables  of  Jupiter. 

TABLE   XVII. 
ECLIPTIC  COORDINATES  OF  JUPITER  FOR  THE  12  POINTS  OF  DIVISION. 


_/"+  Red.  to 

log.  T 

Arg.  i. 

Ecliptic 
(Table  37). 

(Table  60). 

1 

/      a. 

X 

y 

z 

o 

1062.6047 

ota 

93    49    10 

0.716  624 

JOS      I    44 

185    31    36 

-5.1832 

—0.5015 

+0.0137 

I 

I423-6537 

123      o    50 

0.726  798 

134    13    23 

214    43     16 

—4-3816 

—3.0364    +0.0714 

2 

1784.7027 

151      3      o 

0.733  9§2 

162     15    33 

242    45    26 

—24808 

—  4.8184  ,  +0.1114 

3 

2145.7517 

178    26    SS 

0.736  696 

189    39    28 

270      9    21 

+0.0148 

—5-4537       +0.1248 

4 

2506.8007 

205    48    41 

0.734514 

217      I     IS 

297    31      7 

+2.5072 

—4-8125 

+O.IO9I 

5 

2867.8497 

233    44    42 

0.727  800 

244    57    16 

325    27      8 

+4.4010 

—3.0300 

+0.0675 

6 

3228.8987       262    47      7 

0.717865 

273    59    41 

354    29    33 

+5.1982 

—0.5012       +0.0092 

7 

3589.9477    :    293     16    53 

0.707018 

304    29    26 

24    59     19 

+4.6167 

+2.1517        —0.0512 

8 

3950.9967 

325     13    29 

0.608  343 

336    26      2 

56    55    55 

+2.7242 

+4.1841 

—0.0969 

9 

4312.0457    '    358      7      4 

0.694  769 

9    19    38 

89    49    30 

+0.0151 

+4.9518 

—0.1  134 

10 

340.5067 

31      3    59 

0.697  634 

42    16    32 

122      46     24 

-2.6083 

+4.1912 

—0.0948 

ii 

701.5557 

63      9      2 

0.705  860 

74    21     36 

154    Si    28 

-4.5987 

+2.1583 

-0.0474 

TABLE   XVIII. 
JUPITER;  DIRECT  ACTION;  SPECIAL  VALUES  OF  THE  A-COEFFICIENTS  FOR  6  SYSTEMS. 


System  o. 

System  i. 

i 

A 

B 

C 

D 

i 

A 

B 

C 

D 

0 

+0.008  62 

—0.004  22 

—0.004  40 

+0.001  56 

0 

+0.007  65 

—0.003  80 

—0.003  83 

+0.00060 

i 

+0.002  35 

—o.ooi  17 

—0.003  52 

+0.005  25 

I 

+0.003  28 

+0.000  13 

—0.003  40 

+0.004  86 

2 

—  o.ooi  91 

+0.004  50 

—  0.002  59 

+O.OO2  2O 

2 

—  0.001  51 

+0.004  27 

—0.002  76 

+0.002  97 

3 

—o.oo  i  76 

+0.003  72 

—o.ooi  95 

—o.ooi  04 

3 

—  O.O02  13 

+0.004  33 

—  O.OO2  20 

—o.ooo  70 

4 

+0.000  05 

+0.001  53 

—o.ooi  58 

—  O.002  26 

4 

—  o.ooo  13 

+0.001  95 

—  o.ooi  82 

—  O.OO2  52 

5 

+O.OOI  81 

—o.ooo  39 

—  o.ooi  41 

—o.ooi  8  1 

5 

+0.002  06 

—o.ooo  46 

—  O.OOI  6l 

—0.00206 

6 

+0.002  77 

—0.001  37 

—o.ooi  40 

—0.00034 

6 

+0.003  II 

—0.001  55 

—o.ooi  56 

—  o.ooo  19 

7 

+0.002  49 

—0.00095 

—0.001  54 

+0.001  55 

7 

+0.002  43 

—  0.000  76 

—o.ooi  67 

+0.001  93 

8 

+O.OOO  61 

+0.001  30 

—o.ooi  91 

+O.OO2  84 

8 

+O.OOO  IO 

+0.001  87 

—o.ooi  97 

+0.00281 

9 

—  O.OO2  28 

+0.004  86 

—  O.OO2  58 

+0.001  SO 

9 

—  0.002  39 

+0.004  87 

—  O.OO2  48 

+0.00081 

10 

—0.002  O4 

+0.005  63 

—0.003  58 

—0.003  77 

10 

—o.ooi  40 

+0.004  56 

—0.003  15 

—0.00368 

ii 

+0.005  41 

—o.ooo  95 

—0.004  46 

—0.005  89 

II 

+0.004  49 

—  0.00075 

-0.003  73 

—0.004  95 

System  2. 

System  3. 

O 

+0.007  67 

—0.003  8  1 

—0.003  85 

—0.00069 

o 

+0.008  73 

—0.004  27 

—0.004  46 

—  0.001  58 

I 

+0.004  63 

—o.ooo  83 

—0.003  8  1 

+0.005  02 

I 

+0.005  47 

—o.ooo  95 

—0.004  51 

+0.005  97 

2 

—  0.001  42 

+0.004  67 

—0.003  25 

+0.003  80 

2 

—0.002  06 

+0.005  67 

—0.003  60 

+0.003  79 

3 

—  0.002  45 

+0.005  oo 

—0.002  55 

—0.00084 

3 

—  O.OO2  28 

+0.004  86 

—  O.O02  58 

—o.ooi  50 

4 

+O.OOO  12 

+0.001  88 

—  O.OO2  OO 

—  O.OO2  87 

4 

+0.000  61       +0.001  29 

—o.ooi  90 

—  0.002  82 

5 

+O.O02  46 

—o.ooo  78 

—o.ooi  68 

—  o.ooi  92 

5 

+0.002  47 

—o.ooo  94 

—o.ooi  53 

—o.ooi  54 

6 

+0.003  09 

—0.001  54 

—0.001  55 

+O.OOO  21 

6 

+0.002  74 

—o.ooi  36 

—0.001  39 

+0.000  33 

7 

+O.OO2  01 

—0.00043 

—o.ooi  58 

+O.002  04 

7 

+0.001  79 

—o.ooo  39 

—  0.001  40 

+0.001  80 

8 

—  o.ooo  15 

+0.001  92 

—  o.ooi  76 

+O.OO2  46 

8 

+0.00006 

+0.001  52 

—o.ooi  58 

+O.OO2  25 

9 

—  O.OO2  08 

+0.004  23 

—  O.OO2  15 

+0.00068 

9 

—o.ooi  76 

+0.003  7i 

—0.001  95 

+o.ooi  04 

10 

—o.ooi  49 

+0.004  19 

—  O.OO2  70 

—  O.OO2  OO 

10 

—o.ooi  92 

+0.004  52 

—  O.O02  OO 

—  O.OO22I 

ii 

+0.003  18 

+0.000  19 

—0.003  36 

—0.004  82 

II 

+O.OO2  36 

+O.OOI  19 

—0.003  55 

—0.005  30 

COEFFICIENTS  FOR  DIRECT  ACTION. 


8l 


TABLE   XVIII.— Concluded. 
JUPITER  ;  DIRECT  ACTION  ;  SPECIAL  VALUES  OF  THE  A-COEFFICIENTS  FOR  6  SYSTEMS. 


System  4. 

System  5. 

i 

A 

B 

c 

D 

i 

A 

B 

C 

D 

o 

+0.010  47 

—O.OOS  21 

—0.005  26 

—0.00094 

o 

+0.01036 

—0.005  IS 

—O.OO5  21 

+0.001  06 

I 

+0.004  29 

+0.00044 

—0.004  73 

+0.00684 

I 

+0.002  55 

+0.001  57 

—O.OO4  12 

+0.006  16 

2 

—0.002  58 

+0.005  91 

—0.003  33 

+O.OO2  64 

2 

—0.002  34 

+0.005  16 

—  O.OO2  82 

+0.001  95 

3 

—  o.ooi  75 

+0.004  oo 

—0.002  25 

—o.ooi  77 

3 

—  0.001  58 

+0.003  55 

—  0.001  97 

—  0.001  48 

4 

+0.00068 

+0.00096 

—o.ooi  65 

—  O.OO2  46 

4 

+0.00041 

+O.OOI  II 

—o.ooi  51 

—  O.OO2  24 

5 

+0.002  l6 

—  0.00080 

—o.ooi  35 

—  0.001  39 

5 

+0.001  88 

—  0.000  57 

—  0.001  30 

—  0.001  52 

6 

+0.00251 

—o.ooi  25 

—o.ooi  25 

+0.000  14 

6 

+O.002  51 

—  0.001  25 

—  0.001  26 

—  0.000  16 

7 

+0.001  90 

—  0.000  59 

—o.ooi  31 

+0.001  52 

7 

+O.OO2  2O 

—0.00083 

—0.001  37 

+0.001  39 

8 

+O.OOO  43 

+O.OOI  II 

—  o.ooi  54 

+O.OO2  28 

8 

+0.000  72 

+0.00096 

—0.00168 

+0.002  52 

9 

—o.ooi  6  1 

+0.003  62 

—  O.OO2  O2 

+O.OOI  52 

9 

—  0.001  79 

+0.004  10 

—  O.OO2  31 

+O.OOI  82 

10 

—0.002  41 

+0.005  30 

—0.00290 

—  O.O02  OI 

IO 

—  O.O02  64 

+0.00606 

—0.00342 

—  O.OO2  72 

ii 

+0.002  68 

+0.001  55 

—0.004  23 

—0.006  32 

II 

+0.00444 

+0.00036 

—0.004  80 

—0.006  91 

TABLE   XIX. 
JUPITER  ;  DIRECT  ACTION  ;  DEVELOPMENT  OF  THE  A-COEFFICIENTS. 


Arg. 

ufA 

10*  8 

io«  C 

io«Z> 

LJ,  g' 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

O  O 

+1347 

o 

+1237 

O 

-2583 

O 

—  I 

O 

O   I 

—   4 

-58 

—   i 

—  19 

+   4 

+  76 

-  18 

+   3 

1—2 

—   7 

-  77 

+  17 

+  86 

—   12 

—  IO 

+  82 

—  13 

I  —  I 

+  III2 

o 

+  358 

o 

—  1470 

o 

o 

+  539 

I  O 

—  II 

—211 

-  16 

-184 

+   28 

+395 

o 

+   2 

I+I 

—   3 

O 

__     j 

+   2 

+   5 

o 

0 

+   I 

2—3 

—   3 

-  82 

+  8 

+  82 

—   4 

—  i 

+  83 

—   5 

2  —  2 

+3878 

+   2 

-3526 

u  ,   C 

-  355 

+  2 

—  3 

+3699 

2—1 

—   7 

-165 

—   5 

—  S3 

+  12 

+217 

+  53 

—   2 

2   O 

-  16 

+   2 

—  IS 

+  3 

+  30 

—  5 

+  i 

—   II 

o 

O 

o 

o 

0 

o 

o 

O 

3—4 

+  13 

—  40 

—   12 

+  40 

—   3 

o 

+  42 

+   12 

3—3 

+  1779     +  I 

—I7OO 

—  3 

-  79 

+  I 

—  2 

+  1737 

3—2 

—  28 

-672 

+  24 

+S9S 

+   5 

+  77 

+633 

-   25 

3-1 

-  18 

+  3 

-   6 

+  i 

+  24 

—  4 

O 

-  6 

4-5 

+   10 

—  IO 

-   8 

+  10 

0 

+  I 

+  21 

+   9 

4-4 

+  587 

—  3 

—  570 

+  I 

-  18 

+  I 

+   2 

+  575 

4—3 

-  18 

—434 

+  16 

+412 

+    2 

+  22 

+423 

—  17 

4—2 

-  80 

+  7 

+  70 

—  6 

+   H 

0 

-  6 

-  76 

5-6 

+    2 

+  16 

—   3 

—  14 

+    I 

—  I 

+  26 

+  6 

5-5 

+  176 

—   2 

—  172 

+   2 

—   3 

0 

+  3 

+  154 

5-4 

-   8 

-185 

+   7 

+180 

0 

+  5 

+184 

-  8 

5—3 

-  66 

+  6 

+  64 

-  6 

+   4 

—  i 

-  6 

-  67 

82  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

D.   ACTION  OF  SATURN  AND  MERCURY. 

§  40.  The  inequalities  due  to  the  direct  action  of  Saturn  are  so  minute  that  an 
approximate  development  will  suffice.  I  have  therefore  used  the  development  of 
A~3  and  A~5  by  spherical  harmonics.  We  put  a^  for  the  mean  distance  of  Saturn, 
Z,  as  usual,  for  the  difference  of  mean  heliocentric  longitudes  of  the  planet  and 
Earth  (L  —  s  —  g'},  and  a  for  the  ratio  of  the  mean  distances.  With  this  notation 
the  developments  to  a4  are 

J  =  i  +  fa*  +  -Yi6-«4  +  •  •  •  +  (3  +  -V-«2  +  •••)«  cos  L 

+  (¥•  +  -Vg5-"2)*2  cos  2^  +  Y*3  cos  3Z  +  W*4  cos  4Z 

^  o,  i  +  -Y-«2  +  HI40*  +  •  •  •  +  (5  +  JF«2  +  •••)«  cos  Z 

••«*  cos  2Z  +  1-a'  cos    Z  +  1la4  cos    Z 


This  development  is  valid  when  the  eccentricities  are  taken  account  of,  provided 
we  use  the  true  radii  vectores  and  true  longitudes  instead  of  the  mean  ones.  But 
this  is  unnecessary  in  the  present  case.  For  Saturn  we  have 

a  =  0.1070 
Reducing  to  numbers  this  gives 

«• 

s  =  1.0262  +  0.328  cos  L  +  0.044  cos  2-^  +  °-°o5  cos  3^  •  •  • 


TT=  1.0741  +  0.557  cosZ  +  o.no  cos  2Z  +  .016  cos  3Z  +  •  •• 

For  the  geocentric  coordinates  X,  Y,  Z,  of  Saturn  we  have 

X=  a'  —  a,  cos  Z  =  al  (a  —  cos  Z)  Y=  —  al  sin  Z  Z  =  o 

Then 

2^=  ^.(X*  -  Y2)  =  a3^  (a2-  2<z  cos  Z  +  cos  2Z) 

/-         /'^  a3  «i* 

C.  =  «'  C  =  -- 

3   & 

Z>,  =  a'3/?  =  o»|i-  (}  sin  2Z  -  a  sin  Z) 
Reducing  to  numbers,  and  performing  the  necessary  multiplications  we  find 


=  -f  .0027  +  .031  cos  Z  +  .620  cos  2Z 
iosC,  =  —  .419  —  .135  cos  Z  —  .018  cos  2Z 
io3Z>,  =  —  .029  sin  Z  -(-  .607  sin  2Z 


COEFFICIENTS  FOR  DIRECT  ACTION.  83 

Then,  the  principal  terms  are 

io3MK=  +  .0013  +  .015  cos  L  +  .307  cos  zL 
loU/C,  =  —  .208  —  .067  cos  L  —  .009  cos  2Z, 
iosMDl  = —  .014  sin  L  +  .302  sin  2.Z, 

§41.  The  mass  of  Mercury  is  so  minute  that  its  action  upon  Venus,  the  only 
planet  whose  motion  it  can  sensibly  affect,  has  never  been  determined  with  cer- 
tainty. There  is  every  reason  to  believe  that  the  uncertain  determinations  of  the 
mass  which  have  been  made  were  too  great  by  2  or  3  times  their  entire  amount. 
From  Hill's  estimate,  based  on  the  volume  and  probable  density  of  the  planet,  it  is 
very  probable  that  the  mass  is  less  than  i  -=-  10000000  that  of  the  Sun.  From  the 
results  of  §  30  it  is  inferred  that  its  secular  effect  on  the  motion  of  the  lunar  elements 
is  proportionally  yet  smaller  than  its  mass. 

The  only  periodic  inequalities  that  could  become  sensible  are  those  of  compara- 
tively long  period.  Their  probable  limiting  values  are  considered  in  Action, 
p.  273,  from  which  it  appears  that  the  largest  inequality  is  that  depending  on  the 

argument 

/  +  TT  +  T,M'  +  g' 

and  that  the  limiting  value  of  the  coefficient  was  estimated  at  o".i.  For  another 
argument  the  limiting  value  was  o".o4.  These  estimates  rest  on  a  mass  double  of 
what  may  now  be  considered  the  most  probable  value.  For  these  reasons  it  was 
intended  to  leave  the  action  of  Mercury  entirely  out  of  consideration  in  the  present 
investigation.  But,  for  the  sake  of  completeness,  and  to  leave  open  as  few  questions 
as  possible,  it  was  at  length  decided  to  compute  the  action  in  the  same  way  as  that 
of  Venus.  Twelve  systems  and  twelve  indices  were  used.  With  144  special 
values,  it  is  easy  to  compute  not  only  the  secular,  but  the  principal  periodic  terms. 
Among  the  results  are  the  following  constant  terms  and  terms  depending  on  the 
above  argument,  the  form  being 

A  =  A0  +  Ae  cos  (3M'  -|-  £•')  +  A,  sin  (3*1'  +  g') 
A0  =  +  0.867  Ac  =  —  .00059  A.  =  o 

B0—  —  0.381  Bc  =  +  .00035  B.  =  +  -00008 

C0  =  —  0.486  Cc  =  +  .00026  Ct  =  —  .00006 

Z>0  =  -f  0.0022  Dc  =  +  .0005  Dt  =  —  .0023 

K^  ==  +  0.624  Ke  =  —  .00047  Kt  =  —  .00004 

§  42.  K-coefficients.  From  the  preceding  developments  of  A,  B,  C,  and  D  for 
the  four  disturbing  planets  the  coefficients  K  =  yz  (A  —  B)  are  formed,  and 
K,  0^  and  Dt  are  multiplied  by  M. 

This  special  set  of  coefficients,  containing  the  factor  M,  are  designated  as  K- 
coefficients.     Their  values  are  tabulated  for  Venus,  Mars,  and  Jupiter  as  follows. 
The  values  for  Saturn  are  found  at  the  end  of  §  40  preceding. 


82  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

D.   ACTION  OF  SATURN  AND  MERCURY. 

§  40.  The  inequalities  due  to  the  direct  action  of  Saturn  are  so  minute  that  an 
approximate  development  will  suffice.  I  have  therefore  used  the  development  of 
A~3  and  A~5  by  spherical  harmonics.  We  put  a1  for  the  mean  distance  of  Saturn, 
Z,  as  usual,  for  the  difference  of  mean  heliocentric  longitudes  of  the  planet  and 
Earth  (L  =  s  —  g'},  and  a  for  the  ratio  of  the  mean  distances.  With  this  notation 
the  developments  to  a4  are 

J-  =  i  +  fa*  +  *™-a<  +  .  .  .  +  (3  +  _45a2  +  .  .  .)a  CQS  L 

+  (¥•  +  1i<ra*)af  cos  2L  +  -^  cos  3^  +  -3e¥-a4  cos  4Z 
J  -  I  +  -V-«2  +  iff**4  +  .  .  .  +  (s  +  IJia*  +  .  .  .)a  cos  L 

••«'  cos  2L  +  1Aa*  cos  3^  +  JL-a<  cos    Z 


This  development  is  valid  when  the  eccentricities  are  taken  account  of,  provided 
we  use  the  true  radii  vectores  and  true  longitudes  instead  of  the  mean  ones.  But 
this  is  unnecessary  in  the  present  case.  For  Saturn  we  have 

OL  =  O.IO7O 

Reducing  to  numbers  this  gives 

a3 

A  =  i  .0262  +  0.328  cos  L  +  0.044  cos  2jL  +  0.005  cos  3-£  •  •  • 

Ty  =  1.0741  +  0.557  cos  L  +  o.uo  cos  iL  +  .016  cos  3-Z,  +  •  •  • 

For  the  geocentric  coordinates  X,  Y,  Z,  of  Saturn  we  have 

X  '=  a'  —  a,  cos  L  =  al  (a  —  cos  L)  Y=  —  a,  sin  L  Z  =o 


Then 

«'*  a,5 

-  F2)  =  a?-~  (of— 20.  cos  Z  -f  cos  2Z) 


1  3  A 

Z>,  =  «'3Z>  =  of  £i-  (i  sin  2Z  -  a  sin  Z) 

Reducing  to  numbers,  and  performing  the  necessary  multiplications  we  find 

io3/if  =  +  .0027  +  .031  cos  Z  +  .620  cos  2Z 
lo'C1,  =  —  .419  —  .135  cos  Z  —  .018  cos  2Z 
io3D1  =  —  .029  sin  Z  -f  .607  sin  2Z 


COEFFICIENTS  FOR  DIRECT  ACTION.  83 

Then,  the  principal  terms  are 

io3MK=  +  .0013  +  .015  cos  L  +  .307  cos  iL 
loWC,  =  —  .208  —  .067  cos  L  —  .009  cos  2L 
ioWZ>,  = —  .014  sin  L  +  .302  sin  iL 

§41.  The  mass  of  Mercury  is  so  minute  that  its  action  upon  Venus,  the  only 
planet  whose  motion  it  can  sensibly  affect,  has  never  been  determined  with  cer- 
tainty. There  is  every  reason  to  believe  that  the  uncertain  determinations  of  the 
mass  which  have  been  made  were  too  great  by  2  or  3  times  their  entire  amount. 
From  Hill's  estimate,  based  on  the  volume  and  probable  density  of  the  planet,  it  is 
very  probable  that  the  mass  is  less  than  i  -r-  10000000  that  of  the  Sun.  From  the 
results  of  §  30  it  is  inferred  that  its  secular  effect  on  the  motion  of  the  lunar  elements 
is  proportionally  yet  smaller  than  its  mass. 

The  only  periodic  inequalities  that  could  become  sensible  are  those  of  compara- 
tively long  period.  Their  probable  limiting  values  are  considered  in  Action, 
p.  273,  from  which  it  appears  that  the  largest  inequality  is  that  depending  on  the 

argument 

/  +  TT  +  $M'  +  g' 

and  that  the  limiting  value  of  the  coefficient  was  estimated  at  o".i.  For  another 
argument  the  limiting  value  was  o".o4.  These  estimates  rest  on  a  mass  double  of 
what  may  now  be  considered  the  most  probable  value.  For  these  reasons  it  was 
intended  to  leave  the  action  of  Mercury  entirely  out  of  consideration  in  the  present 
investigation.  But,  for  the  sake  of  completeness,  and  to  leave  open  as  few  questions 
as  possible,  it  was  at  length  decided  to  compute  the  action  in  the  same  way  as  that 
of  Venus.  Twelve  systems  and  twelve  indices  were  used.  With  144  special 
values,  it  is  easy  to  compute  not  only  the  secular,  but  the  principal  periodic  terms. 
Among  the  results  are  the  following  constant  terms  and  terms  depending  on  the 
above  argument,  the  form  being 

A  =  A0  +  Ae  cos  (3»f'  +  g')  +  At  sin  (311'  -f  g') 
Aa=  +  0.867  -Ac ~  ~  -oooSP  At  =  o 

B^  =  —  0.381  Be  =  +  .00035  B,  =  +  -00008 

C0  =  —  0.486  Cc  =  +  .00026  Ct  =  —  .00006 

Da=  +  0.0022  Dc  =  -f  .0005  Dt  =  —  .0023 

JfQ=  +  0.624  Ke  =  —  .00047  Kt  =  —  .00004 

§  42.  K-coefficients.  From  the  preceding  developments  of  A,  B,  C,  and  D  for 
the  four  disturbing  planets  the  coefficients  K '=  ^4  (A  —  .£?)  are  formed,  and 
K,  C\,  and  D^  are  multiplied  by  M. 

This  special  set  of  coefficients,  containing  the  factor  M,  are  designated  as  K- 
coefficients.     Their  values  are  tabulated  for  Venus,  Mars,  and  Jupiter  as  follows. 
The  values  for  Saturn  are  found  at  the  end  of  §  40  preceding. 


84 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE   XX. 

K-COEFFICIENTS    FOR    DlRECT    ACTION    OF   VENUS. 


v,   g' 

10s  MKe 

io3  MK, 

%  ios  MCc 

^io3  MC, 

io3  MDc 

io3  MDt 

li 

a 

m 

tl 

o,     o 

+  5-903 

o.oo 

—3.406 

0.000 

o.ooo 

o.ooo 

0,         I 

+  0.44 

—O.I  I 

—0.30 

+0.07 

+0.03 

+0.33 

0,        2 

+  0.13 

—0.14 

—0.09 

+0.10                +0.05 

+0.06 

I,   -2 

+  0.71 

+0.15 

—0.36 

—0.08             +0.08 

—0.52 

I,   —I 

+10.95 

o.oo 

—6.03 

o.oo 

—  O.OI 

—4.92 

I.        0 

+  0.17 

—0.07 

—  O.22 

+0.05 

O.OO 

+O.22 

2,  —4 

+  0.16 

+0.14 

—0.08 

—0.08 

+0.09 

—O.I  I 

2,  —3 

+  0.93 

+0.19 

—0.39 

—0.08 

+0.13 

—0.73 

2,    —2 

+10.75 

o.oo 

—4.99 

o.oo 

—  0.01 

—7.09 

2,    —I 

+  O.O2 

—0.04 

—  0.16 

+0.03 

—O.OI 

+0.22 

3,  -6 

+  O.O2 

+0.03 

—  O.OI 

—O.OI 

+O.O2 

—  O.OI 

3,  -5 

+  O.I7O 

+0.147 

—0.078 

—0.075 

+O.II 

—0.13 

3,  -4 

+  1.08 

+0.22 

-0.38 

—0.08 

+0.17 

—0.91 

3,  -3 

+  10.24 

O.OO 

—4.00 

O.OO 

—  O.OI 

-7.91 

4,  —6 

+  0.18 

+0.15 

—0.07 

—0.07 

+0.1  1 

—0.15 

4,  -5 

+  1.17 

+0.24 

—0.36 

-0.08 

+0.20 

—1.04 

4,  -4 

+  9-42 

o.oo 

—3.14 

o.oo 

—  O.OI 

-7.89 

5,  -8 

+  0.019 

+0.028 

—0.007 

—  O.OI2 

+0.024 

—0.017 

S,  -7 

+  0.19 

+0.14 

—0.06 

—0.06 

+O.I2 

—  0.16 

5,  -6 

+    1.21 

+0.24 

—0.32 

—0.07 

+O.2I 

-1.05 

S,  -5 

+   8.40 

O.OO 

—2.43 

0.00 

—O.OI 

—7-39 

6,  -7 

+  1.19 

+0.24 

—0.28 

—0.06 

+O.2I 

—O.I  I 

6,  -6 

+  7-32 

o.oo 

-1.86 

o.oo 

o.oo 

-6.64 

TABLE   XXI. 

K-COEFFICIENTS    FOR    DlRECT    ACTION    OF    MARS. 


M,      g' 

IO3  MKC 

io3  MX, 

i^lO3  MCC 

y2io>  MC, 

io3  MDc 

io3  MD, 

0,        0 
0,        I 

+  0*0465 

—  O.O2O 

H 

O.OOO 

—0.024 

—  0.1006 
+0.028 

0.000 

+0.029 

O.OOO 

—0.008 

O.OOO 
+O.OIO 

I,  -2 
I,  —I 
I,        O 

—  O.OI2 
+  O.IIO 
—  O.O25 

+0.014 
+0.001 
—  0.029 

+0.017 
—  0.169 
+0.042 

—  0.0  18 

0.000 

+0.045 

—0.008 

o.ooo 

O.OOO 

—O.OI  2 

+0.054 

0.000 

2,  —3 

2,   —2 
2,   -I 
2,        0 

—  O.O02 
+  0.194 

—  0.035 
o.ooo 

o.ooo 

+O.O02 

—0.039 

+O.OI  I 

+0.009 
—0.129 
+0.049 
+0.003 

—  O.OIO 
+0.001 

+0.053 
—  0.016 

+0.006 
o.ooo 
+0.013 

+O.O02 

—  O.OO2 

+0.165 

—0.014 

—  O.002 

3,  —3 

3,  —2 

+  0.233 
—  0.064 

+0.004 
—0.070 

—0.093 

+0.049 

+O.OOI 

+0.053 

—  O.OO2 

+0.050 

+0.217 
—0.048 

4,  —4 
4,  —3 
4,  -2 

+  0.229 

—  0.094 

—  O.OO2 

+0.004 
—0.103 
+0.030 

—0.064 
+0.044 
+0.003 

+0.001 

+0.049 
—  0.028 

—0.003 

+0.000 

—0.019 

+O.22I 

—0.083 
—0.003 

5,  -4 
5,  —3 

5,  -2 

—  0.115 
—  0.005 
+  0.009 

—0.126 
+0.051 
—0.004 

+0.037 
+0.003 
—0.006 

+0.042 
—0.030 
+0.005 

+0.119 
—0.042 
+0.003 

—0.107 
—0.004 
+0.007 

6,  -5 
6,  -4 
6,  -3 

—  O.I22 
—  0.009 
+  O.OI5 

—0.134 

+0.074 
—0.009 

+0.030 

+O.OO2 
—O.007 

+0.034 
—  0.029 
+0.006 

+0.131 
—0.068 
+0.008 

—0.117 
—0.007 

+O.OI  I 

COEFFICIENTS  FOR  DIRECT  ACTION. 
TABLE  XXII. 

K-COEFFICIENTS    FOR    DlRECT    ACTION    OF  JUPITER. 


J,  g1 

io3Ke 

io»AT. 

tfio'AfC. 

^jo3MCt 

ioWZ>c 

ioWZ>. 

lt 

tt 

H 

tt 

II 

// 

0,        0 

+  0.091 

O.OOO 

—2.135 

OJOOO 

O.OOO 

0.000 

0,        I 

—  O.OO2 

—0.032 

+0.003 

+0.063 

—0.030 

+0.005 

I,   —2 

—  O.O2O 

—0.135 

—  O.OIO 

—0.008 

+O.I3S 

—  O.02I 

I.  —I 

+   0.623 

O.OOO 

—  1.215 

O.OOO 

O.OOO 

+0.593 

I,        O 

+  0.004 

—  O.O22 

+0.023 

+0.326 

O.OOO 

+0.003 

I,  +1 

—  O.O02 

—  O.OOI 

+0.004 

o.ooo 

O.OOO 

+O.OO2 

2,  —3 

—  0.009 

—0.135 

—0.003 

—  O.OOI 

+0.137 

—0.008 

2,    —2 

+  6.II9 

+0.006 

—0.293 

+0.001 

—0.005 

+6.114 

2,    -I 

—  O.OOI 

—0.093 

+O.OIO 

+0.179 

+0.087 

—0.003 

2,        0 

—  O.OOI                     —  O.OOI 

+0.025 

—0.004 

+O.OOI 

—  0.018 

3,  —3 

+  2.875 

+0.003 

—0.065 

+O.OOI 

—0.003 

+2.875 

3,  —2 

—  0.043 

—1.048 

+0.004 

+0.064 

+1.046 

—0.041 

CHAPTER   V. 

PLANETARY  COEFFICIENTS   FOR  THE   INDIRECT  ACTION. 

§  43.  Our  next  step  is  to  form  the  coefficients  G,  J,  and  /  which  are  the  planetary 
coefficients  for  the  indirect  action,  and  correspond  to  K,  ^C,  and  D.  These  we  have 
found  to  be  linear  functions  of  the  perturbations  in  the  motion  of  the  Earth  around 
the  Sun  produced  by  the  action  of  all  the  planets.  From  the  way  in  which  they 
are  formed  it  will  be  seen  that  they  should  include  all  deviations  in  the  motion  of 
the  Sun  from  the  actual  formulae  adopted  for  the  expression  of  fl  as  used  in  deter- 
mining the  action  of  the  Sun  itself.  It  would  therefore  be  necessary,  in  strictness, 
to  include  the  effect  of  any  corrections  that  may  be  necessary  to  the  elements  of 
the  Sun's  motion  employed  by  Delaunay.  But  as  the  eccentricity  of  the  Earth's 
orbit  enters  as  a  symbolic  quantity  into  the  theories  of  both  Delaunay  and  Brown, 
it  will  not  be  necessary  to  apply  any  correction  on  this  account.  The  same  remark 
applies  to  the  position  of  the  Earth's  perihelion.  But  as  the  solar  elements  are 
assumed  to  be  constant  in  the  first  integration  it  is  necessary  to  take  into  account 
the  eftects  of  their  secular  variations,  as  well  as  of  the  periodic  inequalities. 

Moreover,  in  developing  the  action  of  the  Sun  upon  the  Moon  for  the  first  inte- 
gration, it  is  assumed  that  the  mean  distance  of  the  Earth's  orbit  is  strictly  connected 
with  its  mean  motion  by  the  fundamental  relation 


It  is  therefore  necessary  to  include  in  8p'  the  constant  correction  arising  from  the 
action  of  the  planets. 

We  may  conveniently  classify  the  various  terms  of  8v'  and  8p  which  are  to  be 
used  in  the  expressions  (60)  as  follows: 

1.  The    terms  arising  from  the  secular   variation    of  the  eccentricity    of  the 
Earth's  orbit. 

2.  Constant  and  periodic  terms  independent  of  the  mean  longitude  of  the  dis- 
turbing planet. 

3.  Periodic  terms  containing  that  mean  longitude. 

§  44.  Secular  terms  arising'  from  the  variation  of  the  eccentricity  of  the 
Earth's  orbit. 

The  action  of  the  Sun  upon  the  Moon  being  a  function  of  the  eccentricity  of  the 
Earth's  orbit  it  follows  that  the  indirect  action  will  vary  with  that  element.  The 
variation  may  be  taken  account  of  by  assigning  to  8v'  and  8p'  the  increments  of  the 

87 


88  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

Earth's  polar  coordinates  due  to  the  variation  of  the  eccentricity.  It  is  not  necessary 
to  take  into  account  the  variation  of  the  solar  perigee,  because  this  element  is 
retained  in  its  general  form  in  the  final  expressions  of  all  the  perturbations. 

To  find  the  required  values  of  oV  and  8p'  we  differentiate  the  expressions  for  v' 
and  p'  in  terms  of  the  eccentricity,  thus  obtaining 

=-7  =  (2  —  %e'  )  sin^'  +  (|-«'  —  ±^-e'  )  sin  2g'  +  *£e'  sin  %g'  +  Vr^'  s'n  4£"' 

(84) 
-  Hk'  —  V***)  cos  zer'  —  yc'2  cos  3^-'  —  14-c'3  cos  $g' 


Putting  Ae'  for  the  increment  of  e'  due  to  secular  variation,  the  values  of  6V  and 
Sp'  to  quantities  of  the  first  order  are  found  by  multiplying  these  derivatives  by  Ae  '. 
To  determine  what  terms  of  higher  order  are  necessary  we  remark  that  for  an 
interval  of  1000  years  before  or  after  1900  we  have 

A*'  =  ±  .000418  =  ±  86".o 
whence 

(A*')2  =  o".o35 

This  quantity  is  so  small  that  the  powers  of  Ae'  above  the  first  order  may  be  dropped. 
But  Ae'  will  contain  terms  in  T2  which  it  will  be  well  to  include  for  the  sake  of 
approximation  to  rigor  in  the  theory. 

Substituting  in  the  values  of  the  differential  coefficients  just  found  the  numerical 
value  of  e'  for  1850, 

e'  =  .0167711 
we  shall  have 

8v' 

5-7  =  1.999  79  sin^-'  -f  .041  92  sin  2g'  +  .000  91  sin  T,g'  -\-  .000  02  sin 

ep> 

,  =  .008  39  —  .999  68  cos^-'  —  .025  15  cos  2g'  —  .000  60  cos  3^-'  —  .000  01  cos  $g' 


The  value  of  Ae'  by  which  these  expressions  are  to  be  multiplied  is  that  used  in 
the  author's  Tables  of  the  Sun: 

Ae'  =  -  8".595  T-  o".026o  T2 

T  being  counted  in  centuries  from  1850. 

The  corresponding  portions  of  G,  J,  and  /are  found  by  substituting  for  oV  and 
Sp'  in  the  expressions  (60)  the  quantities 

e 


COEFFICIENTS  FOR  INDIRECT  ACTION.  89 

If  we  suppose  that  G,  J,  and  /are  expressed  in  the  form 


G=G0 


with  similar  forms  for  J  and  /,  we  find  by  developing  to  e'2 


-   -*  cos  *g  -  i-e  cos 

/,  =  -  K  -  (I  +  i^'")  cos^'  -  ff'  cos  25-'  -  ^'2  cos  3^' 
7i  =  (3  ~  W*'*)  »«»£•'  +  -V-*'  sin  2g'  +  *&e'2  sin  $g' 


The  following  numerical  values  have  not  been  formed  from  these,  but  by  multi- 
plying the  numerical  values  of  the  factors  given  in  (60)  and  (85),  which  are  derived 
from  developments  to  e'  . 

<9,  =  -f  .06238  —  2.24517  cos^'  —  .21296  cos  2g'  —  .01115  cos  3g' 

/,  =  —  .01257  —  0.75039  cosg-'  —  .03771  cos  2g'  —  .00140  cos  3g'  (86) 

/,  =  +  2.96280  sin  g'  +  .21366  sin  2g'  +  .04342  sin  $g' 

§  45.  Terms  independent  of  the  mean  longitude  of  the  disturbing  planet. 
These  terms  arise  from  the  terms  of  Sv  '  and  8/>'  which  are  either  constants,  or  func- 
tions of  g'  alone.  In  the  case  of  the  longitude  the  eccentricity  and  perihelion  of 
the  Earth's  orbit  are  so  adjusted  that  both  the  constant  terms  and  those  dependent 
on  Arg.  g'  shall  vanish,  leaving  the  only  terms  of  8v'  to  be  considered  those  depend- 
ing on  Arg.  2g'  etc.  Both  these  terms  themselves  and  the  factors  by  which  they 
are  subsequently  multiplied  to  form  G,  f,  and  /  are  so  minute  that  the  results  are 
assumed  to  be  insensible;  we  have,  therefore,  only  to  consider  the  terms  of  8p' 
which  remain  after  the  adjustment  of  the  eccentricity  and  perihelion  just  mentioned. 
These  might  be  derived  from  the  numbers  in  Tables  of  the  Sun;  but  the  author 
finds  that  the  results  have  not  been  carried  out  with  the  precision  desirable  in 
the  present  problem.  He  has,  therefore,  computed  these  terms  independently  from 
theory,  using  the  method  of  variation  of  elements,  and  carrying  the  results  to  terms 
of  the  second  order  in  the  eccentricities  and  mutual  inclination.  The  general  for- 
mula are  as  follows.*  The  accented  quantities  refer  to  the  outer  planet. 

Action  of  an  outer  on  an  inner  planet. 

Bp  =  m'a{pa  +  jOj  cos  II  +  (/»0>c  +  ft,,,  cos  II)  cos^  +  ft,,  sin  TI  sing-} 
where 


*The  derivation  has  appeared  in  the  Astronomical  Journal,  vol.  xxv. 


90  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

Action  of  an  inner  on  an  outer  planet. 

V  =  m{pj  +  pj  cos  IF  +  (pa  /  4-  pliC'  cos  IF)  cosg-'  4-  plt,  sin  IF  sing'} 
where 

IF  =  TT'  -  TT 


,    «r 

Pl=2J 

e  e 

Pi  c  — C2-^  +  -^Mi  ft  /  =  ~ 

4  4 

The  two  actions  are  mutually  interchanged  by  replacing  Z>  by  —  (i  +  D)  in 
either.  They  were,  however,  developed  independently  in  order  that  this  relation 
might  serve  as  a  test  of  the  accuracy  of  both. 

The  coefficients  A0  and  A±  are  functions  of  the  mutual  inclination  of  the  orbits 
(<r  =  sin  -j/)  and  of  the  coefficients  b^  defined  by  the  development 

(i  —  20.  cos  L  4  a2)'  =  JS^0  cos  iL 

Dn  means  the  nth  derivative  as  to  log  a,  or  the  symbolic  value  of  [a(5/5a)]n. 
If  it  be  desired  to  use  the  usual  successive  derivatives  as  to  a  itself,  we  may  do  so 
by  the  substitutions 

D  =  «Z>a  Z?3  =  aZ?a  4-  /£D\  IP  =  aDa  +  T><£D\  4-  a?D\ 

From  the  numerical  values  of  the  coefficients  A0,  A^  and  their  Z>'s  we  have  the  fol- 
lowing results: 

Action  of  Venus  iog&p'  =  +  1443.0        +    31  cosg-' —  17  sing-' 

Mars  —      30.          4-    ncosg-'—    Ssing-' 

Jupiter  —1183.1        +    90  cosg-' 4- 50  sin  g-' 

Saturn  —      55.4 

Uranus  —        i.o 


Total  +     173.5         +  132  cos  g-' +  25  sing' 

Additional  to  these  we  have,  for  Mercury,  with  mass  —  io~7 

io9V  =  +  38.0  —  7  cos  g'  +  3  sing-' 
which  I  treat  separately. 


COEFFICIENTS   FOR  INDIRECT  ACTION. 


From  these  (60)  gives  the  following  coefficients  for  G,J,  and  /,  these  quantities 
being  expressed  in  the  torm 

G=G,+  Ge  cos  g'  +  G,  sing' 


Action  of 

io»G0 

109<?c 

io»G. 

lo'/o 

icPJc           io*J,         io9/o     io9/c 

I09/. 

Venus 

+3246 

.8 

+  233-3 

-    38.2 

+  1083.3 

+  77- 

8 

-12.8 

+  1 

•3 

-217.7 

Mars 

-     66 

.8 

+     21.4 

-  18.0 

-      22.3 

+     7- 

2 

-  6.0 

+ 

.6 

+     4-5 

Jupiter 

-2655 

.0 

+  68.8 

+  112.3 

-  886.0 

+     22. 

8 

+  37-5 

3 

•7    —I 

+  178.2 

Saturn 

-   124 

.6 

-     6.3 

o 

—     41.6 

—       2. 

i 

0 

o 

+     8.4 

Uranus 

—           2 

.2 

—       .1 

0 

-         .8 

0 

o 

o 

+          .2 

Total 

+  397 

.6 

+316.8 

+  56.1 

+   132-7 

+  I05- 

5 

+  18.7 

—  i 

.9   -.1 

-  26.6 

Mercury 

85 

.1 

-  11.4 

+  6.7 

+     28.4 

-     3- 

9 

+    2.2 

—  o 

.2        O 

-     5-7 

(87) 


The  totals  here  given  are  not  formed  by  addition,  but  by  an  independent  com- 
putation of  the  entire  amount.  Hence  small  discrepancies  between  the  totals  and 
the  sums. 

§  46.  Periodic  perturbations  of  the  -point  G,  containing  the  mean  longitude 
of  the  disturbed  planet. 

These  are  taken  from  Astronomical  Papers^  Vol.  Ill,  Part  V,  where  they  are 

found : 

For  Venus,  on  pp.  486-488 
For  Mars,  "  "  527-530 
For  Jupiter,  "  "  550-551 

The  perturbations  by  Venus  are  diminished  by  the  factor  .015  for  reduction  to  the 
adopted  value  of  the  mass. 

The  expressions  thus  found  are  shown  in  tabular  form  below.  In  the  original 
the  constituents  of  the  arguments  were  the  mean  anomalies  alone,  but,  in  the  present 
work,  the  longitudes  of  the  disturbing  planets  are  reckoned  from  the  Earth's  perihe- 
lion. In  order  that  the  arguments  for  the  direct  and  indirect  inequalities  may  coin- 
cide, these  perturbations  have  been  transformed  so  that  the  planet's  mean  longitude 
shall  be  reckoned  from  the  Earth's  perihelion.  The  following  are  the  numbers  used: 


Earth 
Venus 
Mars 
Jupiter 


129    27 

333    18 
ii    56 


o°    o' 
29     6 

232    57 
271    35 


Then,  if  any  pair  of  terms  in  8v'  or  8p'  be  represented  by 

v,  cos  (t'gt  +  i'g')  +  v,  sin  (ig^  +  t'g ') 


92 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


we  have  to  compute 


to  transform  them  into 


bc  =  ve  cos  t(-7rt  —  IT')  —  vt  sin  t(irt  —  TT') 
bt  =  vc  sin  t(vt  —  IT')  +  vt  cos  i(ir4  —  TT') 

6,  cos(/74  +  i'g')  +  b,  sm(t7t  +  i'g') 


/4  being  the  mean  longitude  of  the  planet  from  TT',  designated  by  v,  M,  j,  and  s  in 
the  cases  of  the  individual  planets. 

The  original  and  transformed  values  of  &V'  and  8p'  are  shown  in  Tables  XXIII 
to  XXVI. 

The  subsequent  steps  are  shown  in  Tables  XXVII-XXXIV  in  the  following 
order: 

The  values  of  G,J,  and  /  given  in  Tables  XXVII-XXX  are  formed  from  the 
expressions  of  8z>'  and  Sp  in  terms  of  g±  by  the  formulae  (60).  These  are  then  sub- 
jected to  the  transformation  of  §46  and  multiplied  by  the  constant  coefficient 
io3w2  =  5.595.  The  factor  io3  is  introduced  in  order  to  have  the  most  convenient 
unit  in  subsequent  computation.  As  a  check  upon  the  work  the  values  ofy,  G,  and 
/were  also  computed  using  the  transformed  expressions  for  &V  and  Sp',  and  the 
results  compared  with  the  others.  It  has  not  been  deemed  necessary  to  set  forth 
the  steps  of  this  simple  duplicate  computation. 

TABLE   XXIII. 
ACTION  OF  VENUS  ON  THE  EARTH. 


Arg. 

do' 

dp' 

Arg. 

d'j' 

iff 

£"4    g' 

COS 

sin 

COS 

sin 

V,    g' 

cos 

sin 

COS 

sin 

JM 

tl 

// 

u 

u 

// 

// 

a 

-I,     0 

+0.03 

—  O.OI 

—0.041 

—  0.018 

—  I,      0 

o.oo 

—0.07 

-0.045 

+0.004 

—I,    I 

+2.35 

-4-23 

—0.980 

—0.544 

—  i,    i 

—O.OI 

-4.84 

—  I.I2I 

+O.OOI 

—  I,      2 

—  0.06 

—0.03 

+0.032 

—0.006 

—I,     2 

—  006 

O.OO 

+0.025 

—  O.O2I 

-2,      I 

—  O.IO 

+0.06 

+0.041 

+0.065 

—2,      I 

o.oo 

+O.II 

+0.077 

—O.OOI 

—2,      I 

—4.70 

+2.00 

+1.709 

+2.765 

—2,      2 

—  O.O2 

+5.52 

+3.251 

+0.004 

—2,    3 

+1.80 

-1.74 

—0.282 

—0.300 

—2,    3 

—0-53 

-2.45 

—0404 

+0.082 

—2,    4 

+0.03 

—0.03 

+0.018 

+0.016 

—2,    4 

—  O.OI 

—0.05 

+0.023 

—  0.007 

—3,    3 

—0.67 

+0.03 

+O.O2O 

+0.406 

—3,    3 

o.oo 

+0.67 

+0497 

+0.003 

—3,    4 

+I.5I 

—  0.40 

—  0.181 

—0.689 

—3,    4 

—0.33 

-1-53 

—0.697 

+0.149 

-3.    5 

+0.76 

-0.68 

+0.059 

+0.069 

—3.    5 

—0.64 

-0.79 

+0.072 

—0.056 

-3,    6 

+0.01 

—O.OI 

+0.006 

+0.006 

-3,    6 

—  O.OI 

—O.OI 

+0.006 

—0.006 

-4,    4 

—0.19 

—0.09 

—0.079 

+0.160 

—4,    4 

O.OO 

+0.21 

+0.178 

0.000 

—4,    5 

—0.14 

—0.04 

—0.024 

+0.089 

-4,    5 

+O.02 

+0.15 

+0.091 

—  0.018 

—4,    6 

+0.15 

—0.04 

—  O.OI2 

—  0.04  •>, 

-4,    6 

—  O.I  I 

—O.I  I 

—0.034 

+0.030 

-S,    6 

—0.03 

—  O.O2 

—  0.018 

+O.020 

-5,    6 

+O.OI 

+0.04 

+0.026 

—  0.006 

-5,    7 

—  O.I  2 

—0.03 

—  0.018 

+0.065 

-5,    7 

+0.08 

+0.09 

+0.052 

—0.044 

-5,    8 

+0.154 

—  0.001 

o.ooo 

—0.013 

-5,    8 

—0.128 

—0.086 

—0.007 

+O.OI  I 

-8,  ii 

o.ooo 

—  O.OO2 

—  O.OOI 

o.ooo 

-8,  ii 

+0.002 

+0.001 

+O.OOI 

—O.OOI 

-8,    12 

—0.008 

—0.041 

—0.0205 

+0.003  9 

-8,    12 

+0.038 

+0.019 

+0.0093 

—  0.018  7 

-8,  13 

+1.268 

+1416 

+0.004  23 

—0.003  66 

-8,  13 

-1.895 

+0.153 

+0.000  36 

+0.005  58 

-8,  14 

+O.02I 

+0.024 

+0.0106 

+0.004  3 

-8,  14 

—0.032 

+O.O02 

—0.0098 

+0.0058 

COEFFICIENTS  FOR  INDIRECT  ACTION. 


93 


TABLE   XXIV. 
ACTION  OF  MARS  ON  THE  EARTH. 


Arg. 

oV 

w 

Arg. 

<V 

<y 

g<  s1 

cos 

sin 

COS 

sin 

M,  g' 

cos 

sin 

cos 

sin 

I,  —I 

I,        O 

—  0.216 
—0.008 

—0.167 
-0.047 

—0.043 
+0.013 

+0^056 
—0.003 

i,  -I 

I,        0 

—0.003 
—0.033 

+0*273 
+0.034 

+o!o7l 
—O.OIO 

o'.000 
—0.008 

2,  -3 

2,    —2 
2,    —I 
2,        0 

+0.040 

+1.963 
-1.659 

—0.024 

—  O.OIO 

—0.567 
—0.617 
+0.015 

—0.006 
—0.272 
+0.030 
—0.008 

—0.024 
—0.937 
—0.065 

—  O.OI2 

2,  -3 

2,   —2 
2,   —I 
2,        O 

—o.ool 
+0.007 
+1.048 
—0.007 

+0.041 
+2.043 
—1427 
—0.027 

+0.025 

+0.977 
+0.054 
+0.014 

+O.OOI 

—0.005 
+0.047 
—0.005 

3,  -3 
3,  —2 

+0.053 
+0.396 

—  0.118 
—O.IS3 

-0.073 
—0.037 

—0.032 
—0.006 

3,  —3 
3,  -2 

+0.006 
+0.314 

—0.129 
—0.286 

—0.080 
—0.070 

—0.004 
—0.077 

4,  —4 
4,  —3 
4,  —2 

+0.001 

—0.131 
+0.526 

+0.032 
+0.483 
—0.256 

+O.O22 
+O.2I9 
+O.O2I 

—0.008 
+0.059 
+0.045 

4,  -4 
4,  -3 

4,  —2 

+0.008 
+0.366 
-0.582 

—0.033 
-0.342 
—0.059 

—0.023 

-O.I5S 
+0.006 

—  0.005 
—0.165 
—0.049 

S,  -4 
5,  —3 

+0.049 
—0.038 

+0.069 

+O.20O 

+0.041 
+0.041 

—0.029 
+0.008 

S,  -4 
5,  -3 

—0.064 

—  O.2O2 

+0.055 

—  O.02O 

+0.033 
—0.004 

+0.038 
+0.042 

6,  -S 
6,  -4 
6,  -3 

—  O.O2O 
—0.104 
—  O.OII 

—  O.002 
—O.II3 
+O.IOO 

—  O.OOI 
—O.O48 
—0.013 

+0.014 
+0.045 

—  O.OO2 

6,  -5 
6,  -4 
6,  -3 

—  0.016 
—0.153 
+0.059 

+O.OI2 
—O.OI4 
+0.08  1 

+0.008 
—0.006 

—O.OII 

+O.OII 

+0.065 
+0.008 

IS,  -9 
IS,  -8 

+0.018 
+O.2OI 

—0.023 
—0.030 

—  O.OOS 
O.OOO 

—O.007 
—O.OO3 

IS,  —9 
IS,  -8 

—0.027 
—0.083 

—  O.OII 

—0.184 

—0.005 
—0.003 

+0.010 

+O.OOI 

TABLE   XXV. 
ACTION  OF  JUPITER  ON  THE  EARTH. 


Arg. 

3* 

/ 

*/ 

9? 

Arg. 

to 

/ 

<* 

o' 

£"4     g' 

cos 

sin 

COS 

sin 

J,      g' 

cos 

sin 

COS 

sin 

// 

• 

// 

// 

II 

H 

H 

a 

—3 

—0.003 

—O.OOI 

—0.001 

+O.O02 

I,  -3 

—  O.OOI 

+0.003 

+O.OO2 

+O.OOI 

—  2 

—0.155 

—0.052 

—0.037 

+O.O92 

I,   —2 

—0.056 

+0.154 

+O.O9I 

+0.040 

—  I 

—7.208 

+0.059 

+0.026 

+3.3S6 

I,  —I 

—0.140 

+7.207 

+3.356 

+0.067 

O 

—0.307 

-2.582 

+0.108 

—0.042 

I,        0 

-2.589 

+0.236 

—0.039 

—0.109 

+  1 

+0.008 

—0.073 

+0.037 

+0.004 

I,   +1 

—0.073 

—O.OIO 

+0.005 

—0.037 

2     —3 

+O.OII 

+0.068 

+0.049 

-0.008 

2,  -3 

—0.008 

—0.069 

—0.049 

+0.005 

2     —  2 

+0.136 

+2.728 

+1.910 

—0.097 

2,   —2 

+0.014 

—2.731 

—  1.912 

—  0.008 

2     —I 

-0.537 

+1.518 

+0.654 

+0.231 

2,   —I 

+0.619 

—1486 

—0.640 

—0-267 

2,        0 

—  O.022 

—  0.070 

O.OOO 

—0.004 

2,       0 

+0.018 

+0.071 

O.OOO 

+0.004 

3,  —4 

—0.005 

+O.OO2 

+O.OOI 

+0.004 

3,  —4 

—  O.O02 

—0.005 

—0.004 

+O.OOI 

3,  —3 

—0.162 

+0.027 

+O.O2I 

+0.132 

3,  —3 

—0.014 

—  0.164 

—0.134 

+O.OIO 

3,  —2 

+0.071 

+0.551 

+0.378 

—0.049 

3,  —2 

-0.555 

+0025 

+0.018 

+0.381 

3,  -I 

—0.031 

+0.208 

+0.082 

+O.OI2 

3,  -I 

—0.205 

—0.048 

—0.019 

+0.08  1 

94 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE   XXVI. 
ACTION  OF  SATURN  ON  THE  EARTH. 


Arg. 

3v'                        io'<J/>' 

Arg.                          3v'                                   dp' 

g*     g' 

COS 

sin 

cos 

sin 

*,      g' 

COS 

sin 

cos 

sin 

I,  —I 

I,        0 

-0.077 
—  0.003 

+0412 
—  0.320 

+972 

+  18 

+  182 

—      2 

I,  —I 
I,        0 

—0.003 
—0.060 

+0419 
—0.314 

+0.204 
+0.004 

+O.OOI 
—  O.OOI 

2,   —2 
2,    —I 

+0.038 
+0.045 

—  O.IOI 
—O.IO3 

-350 
-236 

-132 
—  IOI 

2,   —2 
2,    —I 

—  O.OOI 

+0.006 

—  0.108 
—0.113 

—0.077 
—0.053 

0.000 
—  O.OO2 

TABLE   XXVII. 
PLANETARY  COEFFICIENTS  FOR  THE  INDIRECT  ACTION  OF  VENUS. 


Arg. 

G 

J 

/ 

gl    g' 

sin 

cos 

sin 

COS 

sin 

cos 

—  I.     0 

—0.190 

—  0^359 

—  0.024 

—0.049 

-0.33 

+o.'i75 

—I,   I 

—  1.220 

—2.203 

—0.408 

—0.735 

—6.34 

+3-52 

—  I.      2 

+0.074 

+0.229 

—  0.014 

+0.005 

—0.125 

—  0.04 

-2,      I 

+0.533 

+0.330 

+O.IOI 

+0.062 

+0.32 

-0.53 

-2,      2 

+6.1  13 

+3-737 

+2.069 

+1.278 

+4.26 

-6.96 

—2,    3 

—0.755 

-0.683 

—0.173 

—  0.180 

-2.63 

+2-73 

—2,    4 

+0.108 

+O.IIO 

+0.007 

+0.009 

—  O.IO 

+0.09 

-3.    3 

+  I.OOI 

+0.014 

+0.359 

+O.OI2 

+0.016 

-0.895 

—3.    4 

—1.590 

—0439 

—0.507 

—0.133 

—0.62 

+2.30 

—3.    5 

+0.191 

+0.143 

+0.039 

+O.O4I 

—i.  02 

+  I-I45 

-3,    6 

+0.056 

+0.051 

+0.005 

+O.005 

—0.045 

+0.05 

—4,    4 

+0.372 

-0.181 

+O.I22 

—0.059 

—0.13 

—0.30 

—4,    5 

+0.100 

—0.057 

+0.069 

—0.019 

—0.06 

—  O.2O 

-4,    6 

—0.099 

—0.026 

—0.030 

—0.009 

—0.06 

+0.23 

-5,    6 

+0.061 

—0.042 

+O.OII 

—0.013 

—0.03 

-0.055 

—5,    7 

+0.136 

—  0.040 

+0.049 

—0.013 

—0.04 

—  0.17 

-5,    8 

—0.031 

O.OOO 

—0.009 

O.OOO 

O.OOO 

+0.231 

—8,    12 

—0.0550 

+0.025 

+O.O02  9O 

—0.015 

—0.007 

+0.036 

-8,  13 
-8.  14 

—0.0093 
+0.073 

+O.OI22 

-0.047 

—0.002  6O 

+0.003 

+0.00298 
+0.008 

+2.126 
+0.093 

+1.903 
+0.079 

COEFFICIENTS   FOR  INDIRECT  ACTION. 


95 


TABLE  XXVIII. 
PLANETARY  COEFFICIENT  FOR  THE  INDIRECT  ACTION  OF  MARS. 


Arg. 

£ 

/                      f 

&<     g' 

sin 

COS 

sin 

COS 

sin 

cos 

I,  —I 

I,        0 

u 
+O.I26 
+O.OIS 

—  o!oo8 
+0.035 

M 

+0.042 

—  O.OOI 

—  o!c>33 
+0.009 

—  0.251 

—  0.073 

—0^324 
—  0.016 

2,  —3 

2,   —2 
2,   —I 
2,        0 

—0.209 
—2.026 
—  O.IOO 

—0.113 

—0.057 
—0.641 
+0.082 
+0.015 

—0.036 
—0.704 
—0.067 

—  O.OIO 

—0.009 
—0.203 
+0.017 
—0.005 

—0.058 
—0.871 
—0.924 
—0.003 

+0.208 
+2.884 
-2485 
—  ox>io 

3,  —3 
3,  —2 

—0.097 
—0.215 

—0.174 
—0.08  1 

—0.026 
—0.072 

—0.056 
—0.029 

-0.186 

—  0.227 

+O.IOI 

+0.594 

4,  —4 
4,  —3 
4,  —2 

—0.009 

+O.III 

+0.097 

+0.085 
+0480 
+0.035 

—  0.005 
+0.045 
+0.035 

+O.O2O 

+0.164 

+O.O2O 

+0.083 
+0.714 
-0.383 

+0.008 
—  0.180 

+0.788 

5,  -4 
S,  -3 

—0.063 
+0.018 

+0.104 
+0.091 

—  O.O22 
+O.O06 

+0.032 
+0.032 

+0.114 
+0.299 

+0.071 
-0.057 

6,  -5 
6,  —4 
6,  -3 

+0.039 

+O.IO2 
—  O.O06 

—  O.OII 

—0.104 
—  0.026 

+O.OI2 

+0.034 

—  O.OOI 

—  O.O02 
—OX»36 
—  O.OII 

—  o.on 
-0.166 
+0.150 

-0.037 
—0.156 
—  0.017 

IS,  -9 
15,  -8 

—0.026 
—O.006 

—  O.O2O 

+0.001 

—0.005 
—0.003 

—O.O06 
O.OOO 

—0.036 
—0.046 

+0.035 
+0.301 

TABLE   XXIX. 

PLANETARY  COEFFICIENTS  FOR  THE  INDIRECT  ACTION  OF  JUPITER. 


Arg. 

G 

/ 

/ 

£"4    g' 

COS 

sin 

COS 

sin 

COS 

sin 

i,  —3 

—0.007 

+0.039 

// 
—  O.O02 

// 
+0.006 

—  o'.039 

—O.OO6 

I,    —2 

—  0.084 

+0-759 

—0.028 

+0.132 

—  0.757 

—0.079 

I,    —I 

—0.064 

+7-554 

+O.O2O 

+2.518 

—10.813 

O.OOO 

I,        0 

+0.238 

-0.266 

+0.082 

+0.032 

—  0479 

-3*70 

I,    +1 

+0.218 

—  O.OI2 

+O.O29 

+O.OO4 

—  0.007 

—0.214 

2,  —3 

+0.360 

—O.O30 

+0.073 

—  0.008 

+  0.026 

+0.352 

2,   —  2 

+4.407 

—0.177 

+  1-447 

—  0.069 

+  0.167 

+4-104 

2,    —I 

+  1437 

+0.522 

+0.527 

+O.I7I 

—  0.808 

+2.231 

2,        0 

—0.039 

—0.023 

+0.013 

+O.OOI 

—  0.036 

-0.097 

3,  —4 

+0.006 

+O.O24 

+O.OOI 

+0.005 

—  0.023 

+0.008 

3,  —3 

+0.096 

+0.290 

+0.023 

+0.008 

—  0.236 

+0.090 

3,  —2 

+0.865 

—O.IO9 

+0.286 

—0.035 

+  0.109 

+0.839 

3,  —I 

+0.177 

+0.028 

+0.069 

+0.008 

—  0.047 

+0.304 

96 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE   XXX. 
PLANETARY  COEFFICIENTS  FOR  THE  INDIRECT  ACTION  OF  SATURN. 


Arg. 

G 

/ 

/ 

*'»    J 

s,  g' 

cos 

sin 

COS 

sin 

COS 

sin 

Ii  —I 
I,        O 

+0-443 

—  O.OOI 

+o"oo5 

—  O.OO2 

+o.'iS3 
+0.007 

+O.OOI 
—  O.OOI 

—0.006 
—0.090 

+0.616 
—0.471 

2,   —2 
2,   —I 

—0.182 
—  0.1  18 

o.ooo 
—0.004 

—0.059 
—0.041 

O.OOO 
—  O.O02 

+O.OOI 

+0.009 

—0.171 
—0.167 

TABLE   XXXI. 

G-COEFFICIENTS    FOR    VENUS. 


Arg. 

itfntG 

lo'w^y 

IO3?«2/ 

V,     g' 

cos 

sin 

cos 

sin 

COS 

sin 

—I,     0 

a 
—  2.272 

a 
+0.048 

—  0.304 

+0.016 

II 

—  0.042 

—  2!o89 

—I,    I 

—14.09 

+0.031 

—  4-703 

+0.005 

—  0.043 

—40.572 

—  I,     2 

+  1.320 

—0.261 

—  0.014 

-0.082 

—  0.536 

—  0.502 

—2,      I 
—2,     2 

+  3-508 
+40.100 

+O.O02 

+0.253 

+  0.663 
+13.764 

+0.003 

+O.I2I 

—  0.071 
—  0.263 

+  3-510 
+45-655 

—2,      3 

—  5.607 

+  1.024 

—  1-354 

+0.346 

—  4-457 

-20.735 

—2,    4 

+  0.838 

—0.205 

+  0.060 

—  O.O22 

—  0.209 

—  0.723 

-3.    3 
—3,    4 

+  5-002 
—  9.004 

+1.186 
+2.036 

+   2.OIO 
-   2.869 

+0.028 
+0.609 

—  0.146 
-  2.855 

+  5.006 
—  13-017 

—3,    5 

+  1.106 

—0.749 

+   0.229 

—0.219 

—  5400 

—  6.668 

*•"        f 

-3,    6 

+  0.326 

—0.270 

+   O.029 

—O.O27 

—  0.239 

—  0.292 

—4.    4 

+  2.314 

—  0.018 

+   0.759 

—O.OOS 

+  0.095 

+  1-827 

—4,    5 

+  1*94 

—  0.180 

+   0.393 

-0.077 

+  0.197 

+  1.152 

^ 
—4,    6 

—  0431 

+0.378 

—  0.128 

+O.I2I 

-  0.873 

—  1.003 

-5,    6 

+  0.389 

—0.148 

+  0.095 

—  O.OIO 

+  0.159 

+  0.312 

—Si    7 

+  0.616 

—0.500 

+  0.215                    —0.185 

+  0.657 

+  0.723 

-5,    8 

—  0.098 

+0.143 

—  0.029                   +0.042 

—  1.065 

—  0.732 

-8,  ii 

+  0.016 

—  O.OI  I 

+  0.005                   —0.006 

—  0.788 

+  1-039 

-8,   12 

+  0.1611 

+0.2094 

+  0.039 

-0.077  7 

—  0.091 

+  0.184 

-8,  13 

+  0.0045 

+0.0661 

+  0.00145 

+O.O22  OS 

—  15-912 

+  1.289 

-8.  14 

-  0.166 

-0.4588 

—  0.040 

+0.024  6 

—  0.681 

+  0.037 

COEFFICIENTS  FOR  INDIRECT  ACTION. 


97 


TABLE  XXXII. 

G-COEFFICIENTS    FOR    MARS. 


Arg. 

io'w2<9 

lO'/M2/ 

io3w2/ 

M,   g' 

cos 

sin 

cos 

sin 

cos 

sin 

I.  —I 

+  0.893 

+O.OI2 

+0.300 

+0.006 

—0^)65 

+  2.293 

I,        0 

—  0.185 

—O.IO5 

—0.035 

—  0.036 

—0.272 

+  0.318 

2,  -3 

+    I.2II 

+0.013 

+0.207 

+0.007 

—0*06 

+  1.208 

2,    —2 

+  12.777 

—0.089 

+4.100 

—0.014 

+0.073 

+17.531 

2,    —I 

+  0413 

+0.595 

+0.335 

+0.194 

+1.161 

+14.821 

2,        O 

—  0.309 

o.ooo 

+0.062 

—  O.OI2 

—0.019 

+  0.129 

3,  —3 

—  1.205 

—0.155 

—0.344 

—  O.O22 

+0.151 

—  1-175 

3,  —2 

-  0.857 

—0.958 

—0.296 

—O.3l8 

+2.641 

-  2.384 

4,  —4 

—  0431 

—0.208 

—0.109 

—0.035 

+0.207 

—  0418 

4,  —3 

—  I-956 

-1-943 

-0.638 

-0.695 

+2.961 

-  2.864 

4,  —2 

+  0.119 

-0.565 

+0.009 

—0.224 

-4.877 

—  0.502 

5,  -4 

+  0404                +0.548 

+0.139 

+0.167 

—0.600 

+  0.452 

5,  -3 

-  0.054 

+0.516 

—0.018 

+0.181 

—I.69S 

—  0.166 

6-4 

—  0.047 

+0.815 

—  O.O2I 

+0.276 

—1.271 

—  o.ioo 

6,  -3 

—  0.130 

+0.073 

—0.05O 

+0.038 

+0495               +  0.685 

IS,  —9 

—  0.109 

+0.147 

—  0.018 

+0.041 

—0.246               —  0.135 

IS,  -8 

—  0.035 

+0.003 

—  0.016 

+0.005 

—0.704 

-  I-55I 

TABLE  XXXIII. 

G-COEFFICIENTS    FOR  JUPITER. 


Arg. 

IO3i 

rfG 

10s* 

•V 

I03 

;»2/ 

J,      g' 

cos 

sin 

cos 

sin 

COS 

sin 

It 

H 

H 

H 

m 

n 

,  —3 

+   0.2  18 

+0.039 

+   O.O22 

+O.OII 

—  0.039 

+  0.217 

,   —2 

+  4.240 

+0-594 

+   0.738 

+0.173 

—  0.553 

+  4.241 

,   -I 

+42.237 

+  1-534 

+  14.088 

+0.274 

—  1.672 

+60471 

,        0 

—  1459 

—1.376 

+  0.196 

-0458 

—21.709 

+  2.097 

,  +1 

—  0.040 

—1.225 

+   0.017 

-0.168 

—  1.203 

+  0.006 

2,  —3 

—  2.OI2 

+0.050 

—   O408 

+O.O22 

—  0.061 

—  I-99I 

2,   —2 

-24.668 

-0.358 

—  8.096 

—  0.061 

+  0.363 

-23488 

2,  —I 

-  7.872 

—3-363 

-   2.887 

—1.  119 

+  5-192 

—12.225 

2,        0 

+  0.218 

+0.140 

—   O.O67 

—  O.OII 

+  0.173 

+  0.542 

3,  —4 

—  O.I4O 

+0.017 

—  0.034 

+0.006 

—  0.017 

—  0.129 

3.  —3 

—    1.677 

+0.403 

-  0.559 

+0.078 

—  0.397 

-  1.365 

3,  —i 

+   O.2I2 

+4£74 

+  0.06  1 

+1.611 

—  4-733 

+   O.2I2 

3.  —I 

—   0.24O 

+0.984 

—  0.078 

+0.380 

—  1.672 

—  O402 

TABLE  XXXIV. 

G-COEFFICIENTS    FOR    SATURN. 


Arg. 

icfufG 

icfm2j 

IO3#/2/ 

s,  g' 

cos 

sin 

cos 

sin 

COS 

sin 

I,  -I 
I,       0 

+248 

+O.OI 

+0.03 
—  O.OI 

+0.86 
+0.03 

// 
+0.01 
+O.OI 

—0.03 
—0.50 

+345 
+2.64 

2,  —2 
2,  —I 

—  I.O2 

-0.66 

o.oo 

—  O.02 

-0.33 
—0.23 

o.oo 

—  O.OI 

0.00 

+0.05 

—0.96 
—0.94 

t 

PART  III. 

FUNCTIONS  OF  THE  COORDINATES 
OF  THE  MOON. 


CHAPTER   VI. 

FORMATION   OF   THE   LUNAR  COEFFICIENTS. 

§  47.  In  attacking  the  problem  before  us  it  has  been  assumed  that  we  have 
expressions  of  the  Moon's  coordinates  relative  to  the  centre  of  the  Earth  as  func- 
tions of  the  six  arbitrary  constants  introduced  through  integrating  the  differential 
equations  in  these  coordinates.  Moreover,  the  constants  in  question  enter  the 
expressions  for  the  disturbing  function  R  only  through  these  coordinates.  It 
follows  from  the  general  expression  of  R  that  if  c  represents  any  one  of  the  six 
lunar  elements,  the  partial  derivatives  of  the  disturbing  function  may  be  derived 
from  the  form 


A  n  ^ 

=  A  -sr  +  B-i:  +  C-.-+  z£>  -..-+•••  (i) 

oc  dc  dc  dc 

It  is  therefore  necessary  to  have  such  expressions  tor  the  squares  and  products  of 
the  coordinates  of  the  Moon  that  each  of  the  required  derivatives  can  be  found  as 
easily  as  may  be. 

When  the  present  work  was  commenced  it  was  intended  to  make  use  of  the 
developments  of  the  powers  and  products  x2,  y1,  etc.,  as  derived  from  Delaunay's 
theory,  and  found  in  Action,  pp.  154-172  and  213-224,  where  the  processes  by 
which  these  quantities  may  be  expressed  are  fully  set  forth.  But,  before  the  work 
was  put  into  final  shape,  Brown's  work  on  the  Lunar  Theory  was  completed  and 
published  so  far  as  the  action  of  the  Sun  was  concerned;  and  it  therefore  became 
a  question  whether  to  use  Brown's  expressions  instead  of  those  of  Delaunay,  or  to 
go  on  with  the  latter.  Each  course  was  found  to  have  its  drawbacks.  The  former 
developments  from  Delaunay's  theory  being  intended  mainly  to  make  an  exhaustive 
search  for  possible  terms  hitherto  unknown  in  the  Moon's  motion,  were  not  com- 
pleted beyond  the  third  order,  though  the  constant  term  was  carried  to  the  sixth 
order.  To  speak  more  exactly,  the  development  was  carried  to  such  a  point  that 
the  square  of  each  coefficient  would  be  correct  to  the  sixth  order. 

It  was  found,  however,  that  the  use  of  Brown's  more  rigorous  theory  would  be 
quite  convenient  except  in  a  single  point.  In  this  theory  the  coordinates  are  explicit 
functions  of  all  the  lunar  elements  except  the  Moon's  distance,  which  enters  into  »/, 
and  of  which  Brown  used  only  the  numerical  value  in  his  developments.  Brown 
has  shown  how  it  is  possible  from  the  data  and  methods  of  his  theory  to  form  the 
complete  derivatives  as  to  this  element  without  using  an  analytic  development  in 
powers  of  m.  But  as  the  application  of  this  method  would  require  a  longer  and 
more  laborious  study  of  the  subject  than  the  author  was  prepared  to  enter  upon,  it 
was  decided  to  use  the  Delaunay  developments  for  obtaining  the  derivatives  as  to 


102  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

>  \  •    • 

log  a.  The  outcome  of  these  considerations  has  been  that,  for  the  sake  of  trial  and 
•3oniparJson,  both  Delaunay's  and  Brown's  developments  have  been  to  a  large  extent 
independently  used,  and  the  results  compared  with  a  view  of  facilitating  an  esti- 
mate of  the  errors  to  which  the  analytic  development  is  subject. 

§  48.  Reduction  of  coordinates  to  the  radius  vector  of  the  Mean  Sun  as 
X-axis. 

In  the  developments  in  Action  the  Sun's  perihelion  was  taken  as  the  origin 
from  which  longitudes  were  measured.  When  the  present  work  was  undertaken,  it 
being  found  that  the  development  of  the  vl-coefficients  would  be  most  easily  effected 
by  taking  the  direction  of  the  mean  Sun  as  the  axis  of  X,  the  same  origin  had  to 
be  taken  for  the  lunar  coordinates.  This  has  been  done  throughout  the  work;  and 
it  must  be  understood  in  the  subsequent  developments  that  x  and  y  are  referred  to 
the  radius  vector  of  the  mean  Sun  as  axis  of  X  except  in  terms  arising  from  the 
motion  of  the  ecliptic. 

In  the  use  of  either  theory  we  'take,  as  the  initial  data  of  the  problem,  the  rec- 
tangular coordinates  of  the  Moon  referred  to  the  mean  Moon  as  the  axis  of  X, 
which  coordinates  we  represent  by  xl  and  y^  These  coordinates  are  those  which 
Brown's  theory  gives  in  the  first  instance,  and  they  are  also  those  which  I  have 
developed  in  powers  of  /«,  etc.,  from  Delaunay's  theory  in  Action,  pp.  167  and  169. 
The  notation  of  arguments  from  the  latter  paper  is: 

gi  the  Moon's  mean  anomaly;  g' ',  the  Sun's  mean  anomaly; 
X,  the  mean  elongation  of  the  Moon  from  the  Moon's  ascending  node, 
equivalent  to  Delaunay's  f,  or  the  mean  argument  of  latitude  ; 
X',  the  same  for  the  mean  Sun  ; 
0,  the  longitude  of  the  Moon's  node; 
/  =  g -\-  IT  -f-  6,  the  Moon's  mean  longitude;   /'  =  g'  +  X'  +  0,  the   Sun's 

mean  longitude  ; 
D  =  /—  /'  =  X  —  X',  as  in  Delaunay,  the  mean  Moon's  departure  from  the 

mean  Sun. 
Putting  Sv  for  the  excess  of  the  true  longitude  over  /,  we  then  have 

xl  =  r  cos  ft  cos  £f  =  aS,k  cos  N 

yl  =  r  cos  ft  sin  &v  =  d2,k'  sin  N  (2) 

z  =  r  sin  ft  =  aLc  sin  N' 

where  k,  £',  and  c  of  dimensions  o  are  developed  in  powers  of  e,  e',y,  and  m. 
The  general  form  of  the  arguments  TV^and  N'  is 

N  or  N'  =  ig  +  i'g'  +j\  +/X'  (3) 

where  g  =  I  -  v,  g'  =  /'  -  TT',  X  =  /  -  0,  X'  =  /'  -  0. 

The  equations  for  transforming  xl  and  yL  into  x  and  y  are 

,v  =  xl  cos  D  —  yl  sin  D  y  =  x1  sin  D  +  yl  cos  D  (4) 


FUNCTIONS  OF  LUNAR  COORDINATES.  103 

If  we  put 


h  =  \(k  +  V)  h'  =  $(k  -  k') 

the  substitution  of  the  development  will  give 


-  =  2A  cos  (D  +  N)  +  2/*'  cos  (D  -  JV)  = 


(5) 
y-  =  2A  sin  (D  +  JV)  +  2A'  sin  (D  -  N}  =  r, 


§  49.  There  are  now  two  ways  of  proceeding  in  order  to  form  the  squares  and 
products.  We  may  either  form  the  last  expressions  for  x  and  y  and  square  them, 
or  we  may  torm  the  squares  and  products  of  x±  and  yly  and  transform  them  from 
the  mean  Sun  to  the  mean  Moon.  Following  the  latter  method  we  have 

x?  =  %(x?  +  y?)  +  \(x?  -  y?)  cos  2D  -  xlyl  sin  2D 

/  =  K*,2  +  y*)  -  \(x?  -  j,2)  cos  2D  +  Xly,  sin  2D 

(6) 
x1  —  J2  =  (x*  —  jKi2)  cos  2D  —  2x^yl  sin 


2xy  =  2xlyl  cos  2D  +  (^2  —  jy,2)  sin 

The  three  junctions  required  in  the  work  being 

(x2  —  y2),         r2  —  3^2         and         2xy 

we  see  from  the  preceding  equations  that  the  first  and  third  can  be  formed  at  once 
from  the  corresponding  functions  of  x*  and  y*  by  a  transformation  through  the  angle 
2D.  If  we  have,  for  any  argument  ^V, 

*,2  —  y?  =  hy  cos  N  2xlyl  =  h2  sin  N  (7) 

the  corresponding  terms  referred  to  the  mean  Sun  are 

**  _  /  =  J(*  +  A)  cos  (iD  +  N)  +  $(*,  -  /y  cos 


(8) 
2xy  =  J(A,  +  /42)  sin  (2Z>  +  ^V)  4-  i(A,  -  A,)  sin  (2.O  -  TV) 


There  are  some  cases  in  which  a  reference  to  a  fixed  axis  is  convenient.    Let  us 

put, 

x0,  y0,  coordinates  referred  to  any  fixed  axis. 

So  long  as  this  axis  is  unrestricted  the  coefficients  for  x0  and  y0  will  be  equal,  as 
is  seen  from  (5).  Hence,  if  we  write  for  any  term  of  x0  and  of  y0  depending  on 
any  argument  N 

xa  =  A,  cos  N  y0  =  At  sin  N  (9) 

this  term  will  be  transformed  into  the  corresponding  term  of  AT  and  of  y,  and  vice 
versa,  by  means  of  the  equations 


104  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

x  =  xt  cos  /'  +  y0  sin  /'  y  =  ya  cos  /'  —  x0  sin  /' 

or 

*•„  =  x  cos  I'  —  y  sin  /'  yg  =  y  cos  /'  +  x  sin  /'  (10) 

The  special  term  (9)  will,  therefore,  transform  into  the  terms  of  x  and  of  y 

l')  (n) 


For  the  special  functions  required  in  the  lunar  theory  we  shall  have  the  follow- 
ing transformations  of  the  same  form  as  (6) 

x2  -  /  =  (*02  -  jr02)  cos  2/'  +  2x0yv  sin  2/' 

(12) 
2xy  =  2x0yQ  cos  2l'  —  (x*  —  jy02)  sin  2/' 

^02  —  y02  =  (x*  —  jK2)  cos  2l'  —  2Ary  sin  2/' 

(13) 
2x0yQ  =  2xy  cos  2!'  +  (x?  —  y2)  sin  2l' 

The  transformation  of  any  one  term  may  be  made  by  the  equations  (6)  by  writ- 
ing +  2/'  or  —  2/'  for  2D. 

If,  as  in  most  of  the  present  work,  the  solar  perigee  is  taken  as  the  fundamental 
fixed  Jf-axis,  we  write  g'  instead  of/'  in  these  equations. 

An  important  remark  to  be  made  on  these  transformations  of  terms  from 
one  axis  to  the  other  is  that  the  equality  of  coefficients  expressed  in  the  equations  (9) 
and  (n)  is  true  only  when  the  fixed  axis  of  J^is  unrestricted.  If,  as  will  sometimes 
be  more  convenient,  we  take  the  direction  of  the  so'ar  perigee  for  this  axis,  some 
values  of  argument  N  in  (9)  will  be  equal  with  opposite  signs.  By  combining  the 
terms  depending  on  these  arguments  the  equality  in  question  will  cease  to  hold. 
If,  however,  the  Sun's  eccentricity  is  dropped,  the  general  equations  will  remain 
valid  for  the  Sun's  perigee  also. 

It  thus  happens  that,  in  the  developments  given  in  Action,  pages  213-215,  the 
coordinates  are  quite  general,  while  the  expressions  for  their  squares  and  products 
given  on  pp.  217-223  are  not  general,  because  the  solar  perigee  is  here  taken  as  the 
fundamental  axis. 

§  50.  Recalling  that  throughout  the  work  we  use  the  symbol  D  to  represent  the 
logarithmic  derivative  as  to  a  of  any  function,  a  serious  question  is  that  of  determin- 
ing the  value  of  this  derivative  with  the  necessary  precision  in  each  special  class 
of  terms.  In  actually  performing  the  work  so  many  tentative  combinations  have 
been  made,  as  better  and  better  methods  were  found,  that  it  is  difficult  to  present 
any  one  process  as  the  definitive  one.  The  following  method  was  at  length  seen 
to  be  the  best  under  the  circumstances  I  have  described.  Let 

u  =  a'<f>(»i) 
be  any  function  of  the  coordinates  of  which  D  is  to  be  formed.     Practically  i  will 


FUNCTIONS  OF  LUNAR  COORDINATES. 

be  equal  to  i  or  2  according  as  the  expression  we  are  dealing  with  is  of  the  first  or 
second  degree  in  the  rectangular  coordinates.  If  we  can  compute  the  value  of 
D(j)(m}  with  sufficient  precision  the  complete  value  of  Du  will  be 

Du  =  /«*<£(»*)  -f  alD<f>(m)  (14) 

If  it  is  developed  in  powers  of  m,  <f>(tn)  =  OQ  +  aim  +  <V«2  +  •  •  •  and  we  shall  have 


£><j>(m)  =  f  otjW  +  3<x27»2  +  •  •  • 

the  coefficient  of  each  term  being  f  of  the  exponent  of  m.     (v.  §12,  Eq.  23.) 

If  we  have  the  numerically  accurate  value  of  any  tf>(m)  from  Brown's  theory 
and  an  approximate  one  from  the  analytic  development,  the  comparison  of  the  two 
will  furnish  a  rude  index  to  the  probable  value  of  the  omitted  powers  of  m  in  the 
development.  It  follows  that  the  nearest  approximation  to  the  value  of  Du  will 
be  obtained  by  using  in  the  first  term  of  the  second  member  of  (14)  the  numerical 
value  of  a{(j)(m)  =  u,  the  analytic  development  being  used  only  for  the  second  term. 
Moreover,  having  an  approximate  estimate  of  the  value  of  the  omitted  terms  of  the 
analytic  development  of  the  second  member  of  (14),  we  may  use  it  to  correct  the 
last  term  of  this  member.  I  conceive  that  no  lack  of  theoretical  rigor  pertaining 
to  this  process  will  lead  to  an  error  of  the  slightest  importance  in  the  present  work. 

§  51.  Formation  of  the  D's  from  Delaunay's  Theory. 

In  the  final  formation  of  the  /^-derivatives  I  have  extended  the  developments 
given  in  Act  to*,  by  the  aid  of  Delaunay's  results,  as  follows.  Delaunay  expresses 
the  reciprocal  of  the  Moon's  radius  vector  in  a  form  which  we  may  write 


a 
- 


where  tt\,  is  put  for  the  sum  of  an  infinite'  series  of  terms,  each  developed  in  powers 
of  ?«,  as  well  as  of  e,  e'  ',  and  y.  This  quantity  77^  is  related  to  the  Moon's  parallax  TT 
by  the  equation 

sin  TT  =  —  (i  -f  TT.) 
a  ^ 

#!  being  the  Earth's  equatorial  radius. 

It  is  to  be  remarked  that  Delaunay's  expression  for  the  parallax  was  only  carried 
to  terms  of  the  fifth  order,  so  that  it  does  not  suffice  for  all  theoretical  purposes. 
It  is  indeed  fairly  probable  that  it  would  suffice  for  the  object  now  in  view.  In 
order,  however,  to  lessen  the  danger  of  any  insufficiency  in  this  respect  I  have,  in 
forming  the  value  of  TT^  compared  each  coefficient  in  the  expression  of  Delaunay's 
parallax  found  in  my  transformation  of  Hansen's  lunar  theory  with  the  more  accu- 
rate value  derived  from  Hansen's  or  Brown's  expression.  We  may  conceive  that 
the  correction  necessary  to  reduce  Delaunay's  coefficient  to  Hansen's  value  is  of 
the  form 

STT,  =  ajn*  +  «<+lw'+1  -\  ---- 


106  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

in  which  i  is  the  power  of  m  next  above  the  highest  to  which  Delaunay  has  carried 
his  coefficient.  From  what  we  have  already  shown  it  follows  that  the  corresponding 
correction  to  Dtr^  is 

an  approximate  value  of  which  is 

In  order  to  make  this  correction  rigorously  exact  we  should  know  the  values  of 
the  coefficients  of  the  omitted  powers  of  i.  This  being  unknown,  the  minute  cor- 
rection is  to  a  certain  extent  a  matter  of  estimation.  I  do  not  conceive,  however, 
that  the  uncertainty  is  at  all  important  in  the  present  investigation. 

We  have  next  to  consider  the  Z>-derivatives  of  the  three  functions 

p2  —  3?2;   fi*  —  '"li  >    and   2fi7?i 

Starting  with  the  equations  (2)  the  values  of  8v  and  ft,  developed  in  powers  of  m, 
are  given  at  the  end  of  Delaunay's  TAeorie,  Vol.  II.  The  values  of  D§v  are 
formed  from  these  with  great  lacility  by  means  of  the  form  (23)  of  §  12,  because 
Delaunay  gives  the  numerical  value  of  each  part  of  every  term  of  the  longitude. 

The  steps  of  the  subsequent  process  consist  in  simple  trigonometric  multiplica- 
tions, and  are  presented  in  tabular  form  on  the  following  pages.  The  fundamental 

quantities  are 

a  a 

7T.  = i,    Sz',    ft         and         D  -,    Dov.    Dfj 

r  r 

which  are  formed  from  Delaunay's  numbers  in  the  way  just  shown. 

The  following  functions  are  then  formed  by  trigonometric  multiplication 

P  =   I   —  TTj  +  7T,2  —  7T,3  +   •  •  •        p~  =   I   —  27T,  +   37T,2  —  4^,'  +    •  •  •         p*=  I  —377-,  +  67T,2—  IO7r,3+   •  •  • 

In  the  final  work,  however,  p  has  been  formed  from  Brown's  theory.     Then 

p>  =  ?  +  r,*  +  t;* 


2\/3'  Cos2/3  =  i  -  Sin2# 

sinSz;  =  8»-|8^  cos  &v  =  i  -  iSz;2 

%?  —  V?  —  /»*  cos2  /8(i  —  2  sin2  8z>)         ^,17,  =  p*  cos2  ft  cos  Sv  sin  Sv 

S=ps'mft  ?2  =  /)2sin2y3  D •  p2  =  -  2p^£hrl 

£>  sin  0  =  cos  ftDft      D  sin2  ft  =  2  sin  ftD  sin  ft      Z>p2  cos*  ft  =  cos2  ft  Dp2  +  pW  cos2  ft  (16) 
Z7  sin2  Sz>  =  2  sin  Sz^Z?  sin  Sy  =  2  sin  8>v  cos  SvDSv 
%?  -  *)*)  =  (i  -  2  sin2  fo) Dp-  cos2  /9  -  2/>2  cos2  ftZ>  sin2  «w 

?  •  Stfi  =  sin  Sy  cos  BvDp2  cos2  y3  +  p2  cos2  /9(i  —  2  sin2  Sv)2)&v 
D  C  =  p*D  sin2  ft  +  sin2  ftZ)  •  p2 


FUNCTIONS  OF  LUNAR  COORDINATES.  107 

The  same  method  might  be  used  to  form  the  derivatives  as  to  the  e  and  y,  but 
this  has  been  deemed  unnecessary,  as  they  can  be  formed  with  entire  precision 
from  Brown's  Theory,  and  probably  with  all  necessary  precision  from  the  develop- 
ments found  in  Action  with  some  extensions  in  special  cases.  As  a  matter  of 
fact  they  have  been  formed  by  both  methods. 

§  52.  Derivatives  from  Brown's  theory.  To  form  the  partial  derivatives  as  to 
Delaunay's  e  and  y  from  Brown's  expressions  it  is  to  be  noted  that  Brown  uses 
instead  of  e  and  y  two  constants  e  and  k  which,  omitting  unimportant  terms,  are 
expressed  thus  in  terms  of  the  Delaunay  elements: 

e  =  (2.000543  +  ,o^en)c  —  .3668^  —  2.oi2ey* 
k  =  (1.000128  —  .O004e'2)7  —  .4967*  —  0.499^7 

A  distinction  is  to  be  made  between  the  a  of  the  present  work,  defined  by  the  con- 
dition a3n?  =  p,  and  Brown's  a,  used  in  his  work.  Brown's  e  is  defined  as  the 
coefficient  of  sin  g  in  the  development  of  jj/a,  or,  using  the  notation  of  the  present 

paper,  in  the  development  of 

a     r  .  a 

—  •  —  cos  p  sin  0v  =  —  i} 
a     a  a 

This  will  enable  us  to  make  a  comparison  of  the  preceding  value  of  e  with  that 
to  be  derived  from  the  analytic  development  in  Action,  p.  168,  from  which  we  find 

-  e  =  (2  -  ^«'  +  ^ '«'  +  *firW>«4  +  V»'«'>  -  (|  +  i72  -  T\V»V  -  (2 


Brown's  2k  is  the  coefficient  of  sin  X,  X  being  the  mean  argument  of  latitude,  in 
the  development  of  r/a  sin  ft,  tound  on  p.  159  of  Action.  From  the  coefficient  as 
developed  in  Action  we  find 

a  2 

0 

Brown  also  gives 

^=.999093;  ^=1.000908 

The  two  results  are  as  follows,  B  indicating  those  from  Brown's  formulae,  A  those 
Irom  the  analytic  development. 

B;  e  =  2.000557^  —  .367^  —  2.oi2^y2  B ;  k  =  1.0001287  —  .499^7  —  .4967* 

A  ;  e  =  2.000426^  —  .371^  —  2.OO4C72  A  ;  k  =  1.0001087  —  .501^7  —  .5007* 

The  difference,  arising  from  the  dropping  of  higher  powers  of  m  in  the  analytic 
development,  is  too  small  to  affect  the  solution  of  our  present  problem. 

To  find  the  partial  derivatives  of  any  function  u  of  e  and  k  with  respect  to  e  and 
y  we  have  from  the  preceding  expressions 


io8  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

du      dude      dudVi  du  du 

d~e=dede  +  dkfo  =  I'9932fe~  •°°99d]i 

(17) 

du      du  de      du  5k  „  Bu  du 

a~  =  5-=-  +  5r=-=  o.ggSo  5l  —  .0025;  -5- 
dy      de  dy  T  3k  £7  /yo    3k  a  de 


These  equations  enable  us  to  find  the  derivatives  as  to  e  and  y  from  Brown's  as  to 
e  and  k. 

§  53.    Tables  of  the  functions  and  derivatives  of  the  Moon's  coordinates. 

The  numerical  processes  by  which  the  required  functions  of  the  coordinates 
were  developed  may  be  followed  and  tested  by  the  aid  of  the  following  tables. 
The  notation  of  the  arguments,  expressed  by  the  indices  in  the  first  column,  has 
been  defined  in  §  48. 

Owing  to  the  circumstances  mentioned  in  §  46,  and  to  the  widely  different  degree 
of  precision  required  in  the  coefficients  of  different  arguments,  the  numbers  of  these 
tables  are  not  always  consistently  continuous.  The  terms  of  many,  perhaps  more 
than  half  the  arguments,  lead  to  no  sensible  inequalities;  with  these  pains  were  not 
always  taken  to  reach  a  higher  degree  of  precision  than  was  required  to  show  the 
order  of  magnitude  of  the  results.  In  the  preliminary  steps  of  the  investigation  it 
was  deemed  sufficient  to  carry  the  expressions  for  the  Moon's  coordinates  to  the 
5th  place  of  decimals,  and  those  for  the  derivatives  to  the  3d  or  4th  place.  But 
when  the  inequalities  of  the  elements  themselves  were  reached  by  integration,  it 
was  found  that  this  degree  of  precision,  while  more  than  sufficient  for  the  periodic 
terms  in  general,  was  not  sufficient  either  in  the  terms  related  to  the  evection,  or 
in  those  determining  the  secular  variations  and  accelerations  of  /,  IT,  and  6.  A 
number  of  successive  revisions  was  found  to  be  necessary,  in  which  the  coefficients 
depending  on  the  argument  g'  were  carried  to  the  yth  place  of  decimals.  As  the 
last  place  was  always  more  or  less  doubtful  only  the  sixth  place  has  been  included 
in  the  printing. 

It  may  also  be  remarked  that  in  commencing  the  tables  it  was  supposed  that  the 
analytic  development  in  Action  would  suffice  for  the  work.  This  expectation 
proving  ill-founded,  the  developments  of  the  Moon's  coordinates  given  by  De- 
launay,  then  those  by  Hansen,  and  finally  those  by  Brown  were  successively 
used  in  the  case  of  those  terms  in  which  greater  precision  was  needed.  Finally 
the  /^-derivatives  were,  in  their  important  terms,  recomputed  by  formula;  proposed 
by  Dr.  Ross,  which  were  much  briefer  than  those  already  given  in  (16)  of  §  51. 

The  want  of  homogeneity  thus  arising  in  the  tables  could  be  cured  by  a  fresh 
development  from  the  fundamental  data  of  Brown  and  Delaunay,  but  I  do  not  think 
any  important  change  would  thus  result  in  the  expressions  for  the  inequalities  of 
the  Moon's  elements  found  in  Part  IV. 


FUNCTIONS  OF  LUNAR  COORDINATES. 

TABLE  XXXV. 

FUNCTIONS  OF  THE   LATITUDE  AND  THEIR  DERIVATIVES. 


109 


Arg. 

ft 

4¥ 

sin/9 

Z>sin/9 

C=/>  sin  ft 

z»c 

dflde 

W 

****** 

sin 

sin 

sin 

sin 

sin 

sin 

sin 

sin 

—  2       O     I        O 
—  I        O     I        O 
O       O              O 
I        O              O 
2O             O 

—.000154 
—.004842 
+.089  503 
+.004898 
-j-.ooo  301 

+.00009 
+.00032 
.00000 
—.00005 
—  .000  01 

—.000  154 
—.004  840 
+.089413 
+.004896 
+  000301 

+.OOOOO 
+.00032 
.OOOOO 
—.00005 

—.000088 
—.007  240 
+.089474 

+.002  466 

+.00008 
+.00044 
—.00018 
+.00003 

—.0049 
—.1304 
—.0049 
+•0454 

—0.0030 
—  0.161  5 
+1.998 
+0.055 

O  —  I               O 
I   —  I               O 

+.000023 

—.000038 

+.000023 

—.OOO  038 

+.OOOO29 

—.000  017 

+.000052 

+O.000642 

—  I   —  I               O 

O        I               O 
0030 

—.000  032 

—  .000  030 

+.000013 

+.OOO  OI 

—.000  030 
—  ooo  030 

+.000013 
+  OOO 

—.OOOO24 

+.000031 

—.000079 

—0.000  535 

—I     03—2 

—  I    —  I      ^   —  2 

+.000968 

+.OO2O2 

+.000968 

+.O02  OI 

+.000545 

+.00124 

+.0097 

+O.OI2  2 

O       03  —  2 

+  000568 

+  ooi  80 

+  000568 

+  001  80 

_L  nm  58 

—  I   —  I      I   —  2 

O  —  I      I   —  2 

+.000  144 

+.OOO3O 

•4-.OOO  idd 

+  ooo  30 

+.000  165 

+  ooo  18 

—  I        O     I   —2 
O       O     I   —  2 
I        O     I   —  2 
Oil    —  2 

+.000808 

+.OO3  O22 

+.000  161 
—  .000060 

+.001  75 
+.00542 
+.00028 

—  .OOO  12 

+.000808 
+.003  020 
+.000  161 
—  000060 

+.001  74 
+.005  40 
+.00028 

—  .OOO  12 

+.001  179 
+.003308 
+.000080 

—  .000061 

+.00244 
+.00647 
+.OOOI2 

+.000  18 

+.02  1  i 
—.0007 

4-0.0261 
+0.0739 
+0.0017 

TABLE   XXXVI. 
FUNCTIONS  OF  THE  RADIUS  VECTOR  AND  LATITUDE. 


Arg. 

DK, 

/)2=r2/a2 

c 

f>*  cos2/3 

D.  ff 

D.C 

D.  f>2cos*p 

g  g*  I   * 

cos 

cos 

COS 

COS 

COS 

COS 

COS 

O       0     O       O 

+.002  638 

+1.002866 

+.004  038 

+.908828 

-.00443 

+.OOOOI 

-.00444 

I        O     O       O 

—.OOI40 

-0.10858 

—.00042 

—.108  17 

+.00480 

+.00006 

+.00474 

2       O     O       O 

—.OOOl6 

—  0.001  52 

—.00002 

—.001  51 

+.00008 

+.OOOOI 

+.00008 

—  I        I     O       0 

+.000917 

—0.00070 

—.OOOOO 

—.00070 

—.ooi  77 

.OOOOO 

—.00176 

O        I     O       O 

—.OO0267 

+0.000  266 

+.OOOOOI 

+.000266 

+.000  713 

+.000001 

+.000712 

I         I      O        O 

—.000  637 

+0.000  54 

.OOOOO 

+.00054 

+.001  27 

.OOOOO 

+.001  26 

—  I        O     2       O 

+.OOOIO 

+0.000  32 

+.00065 

—.00033 

—.000  19 

—.00003 

—.00015 

O       O     2        O 

+.00006 

—  O.OOOO2 

—.00398 

+.003  97 

—.00010 

+.000  O2 

—.OOO  12 

1020 

+.OOOOO 

+O.OOOOO 

—.00022 

+.OO022 

.OOOOO 

.OOOOO 

+.OOOOI 

—  2        O     2   —  2 

—.00032 

+0.001  83 

.OOOOO 

+.001  82 

+.00380 

+.OOOO2 

+.00377 

—  I        O      2   —  2 

+.019  66 

—0.018  91 

—.00003 

—.01888 

—.035  22 

—.00006 

-.035  15 

O        O     2   —  2 

+.02643 

—0.014  86 

—.00029 

-.014  57 

—.049  46 

—.00054 

—.04892 

I        O2    —2 

+.002  91 

—o.ooo  48 

—.00002 

—.00046 

—.00154 

—.00003 

—.001  50 

2        O     2   —  2 

+.00027 

—0.00004 

.OOOOO 

—.00004 

—.00007 

.OOOOO 

—.00007 

—I         I      2    —2 

+.000216 

+0.00006 

.OOOOO 

+.00006 

—.00048 

.OOOOO 

-.00047 

O                2   —2 

—  .000441 

+0.000  144 

+.000005 

+.000  139 

+.00084 

—  .000016 

+.000856 

I                2   —2 

—.000  058 

0.00000 

.OOOOO 

.OOOOO 

+.OOOOI 

.OOOOO 

+.OOOOI 

—  I    —        2    —2 

+.000844 

—0.00080 

.OOOOO 

—  .00080 

—.001  40 

.OOOOO 

—.OOI  40 

O   —        2    —  2 

+.OO2  O27 

—  o.ooi  043 

—  .000015 

—.001  028 

-.00386 

—.00002 

—.00384 

I    —        2   —  2 

+.000260 

O.OOOOO 

.OOOOO 

.OOOOO 

—.00015 

.OOOOO 

—.00015 

—  I        O     O        2 

—.00002 

—0.000  05 

—.00002 

—.00003 

—  .00002 

—.00004 

+.OOOOI 

0002 

—.00008 

+0.0005 

+.000287 

—.00023 

+.00016 

+.00056 

—  .00040 

1002 

—.00004 

+0.0006 

+  .OOOII 

—.00005 

—.00007 

+.00023 

—.00030 

O        2      O        O 

—  .000  018 

+.000  036 

+.000  036 

O   —  2     2   —  2 

+.000  106 

—  .OOO  212 

—  .OOO  212 

O        22   —  2 

—  .000  004 

+.000  008 

+.000008 

no 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE   XXXVII. 
FUNCTIONS  OF  THE  LONGITUDE  AND  THEIR  DERIVATIVES. 


Arg. 

sin  Sv 

cos  Sv 

isin  2<?Z>, 

sin2  dv 

D.tiifdv 

Dto 

\D  sin  2<Jz> 

g  g*  it 

sin 

COS 

sin 

cos 

cos 

sin 

sin 

-4-  <v/i8i 

I       O     O       O 
2000 
—  I        I     O       O 
O        I     O       O 
I        I     O       O 
—  I        O     2       O 
O       O     2       O 

+.IOQ58 
+00371 
—.00072 
—.003  233 
—.00053 
—.00019 
—.00199 

—.00035 
+.00300 
+.00017 
—.00002 
—.00018 

+XXXJII 
—  .00001 

+.10903 
+.00365 
—.00071 
—.00320 
—.00053 
—.00019 
—.001  97 

-(-.OUO33 
+XWO7O 
—.00597 
+.00033 
+.00004 
+.00036 
—  .00022 
+.000  02 

+.001  55 
+.000  20 
—.000  19 
+.000  249 
+.00036 
+.00003 

—  XXX)  OI 

.000000 

—.000138 
—  .002  052 

—.002  8437 

—.001  199 
+.000215 
+.000091 

—.OOO  320 
—.000314 
—  .002008 
—.002  7902 
—.OOI  2OO 
+.OOO2I2 
+.OOOOOO 

—2       O     2  —2 
—  I        O     2  —2 
0       O     2  —2 
I        O     2  —2 
2        O     2  —  2 
—  I               2  —  2 
O               2  —2 
I               2  —  2 

+.001  03 

+.O22  17 
+.OII45 
+.OOO96 
+.00009 
—  OOOI4 
—.OOOI2 

—  JOOI  25 

—  .00058 
+.001  17 
+.00067 
+.00007 

—XXX)  03 

—  .00003 

+.001  O7 

+.02203 
+.01135 
+  00008 

+.000  15 
—  .000  14 
—.00012 

+.00249 
+.001  16 
—.00234 
—.00134 
—.00015 
+.00007 
+.00006 

+.00531 

+.003  77 
—  .00483 
—.00427 
—.00048 
+.000  20 
+.000148 
+.00008 

+.002  545 

+.047405 
+.037838 

+.003  056 
+.000228 
+.000482 
—.0003608 

+.OO2  7O5 
+.046777 

+•037  341 
+.003  273 
+.000469 
+.000400 
—.000  3662 
—  xxx)  075 

—  I   —        2  —2 
O  —        2—2 
I   —        2  —  2 
—  I       O     O       2 

+.OOIOO 

+.00080 
+.00007 

—.0000  1 
+.00007 
+.00005 

+.00009 
+.00080 
+.00008 

+.000  02 
—.00016 

—  .000  IO 

+.00007 
—.000451 
—.00039 

+.O02  274 

+.002  9357 
+.000271 

+.002  223 

+.0020079 

+.000305 

0       O     O       2 
I        O     O        2 

—  .00027 

-J-.OOOOI 

—.00027 

—  .00002 

—.OOOO8 

—.000204 

—  .000300 
+  ooo  008 

O       2     O       O 

ooo  036 

O  —2     2  —  2 

+  000039 

TABLE   XXXVIII. 

COORDINATES  REFERRED  TO  MEAN  MOON  AND  THEIR  DERIVATIVES. 


Arg. 

f, 

Brown. 

z*t 

Anal. 

%J3e 

Brown. 

*Ar 

Brown. 

f, 

Brown. 

A, 

Anal. 

fyjde 
Brown. 

W*T 

Brown. 

g  g'  i  v 

cos 

cos 

cos 

cos 

sin 

sin 

sin 

sin 

O       O     O       O 

+•995  47 

—  .00302 

—  0^8  £ 

080/4 

I        O     O       O 

—  JOU  4O 

-L-  OOI  27 

One  e 

2       O     O       O 

+.001  51 

+.OOOOI 

•4-  OSS  01 

—  I         I      O        O 

—  .000  174 

+.000  04 

—  OOt  17 

O        I     O       O 
I        I     O       O 

+xxx>iO34 

-KOODOOS 

+.00033 
—  00004 

—.000346 

—xxx)  063 

—.003  2283 

—  .003  12 

—  O.OO2  508 

+.OOI5II 

—  I        O     2       O 
O       O     2       O 

+.00004 
+.001  09 

.00000 
+.OOO  01 

+.0008 

+xx>i8 

4-  nnS  i 

—.00025 

.00000 

—0.004  4 

—.OIO9 

OO78 

I        O     2       O 

—  2        O     2  —  2 

+.00005 
—  .000  so^ 

.00000 
—  ooo  63 

+.OOIO 

+.0025 

—  .OOOO5 

XXX)  OO 

—  O.OOIO 

—.OO25 

—  I        O     2   —  2 
O        O     2   —  2 
I        O2   —  2 

—.010  16 

—XXX)  36 

+xx»i8 

—  x>i93i 
—.022  85 
+.00029 

-.1844 
+.0237 
+.0048 

+.0019 
+XXX58 

+.022  24 
-j-  .OIO  32 

+XH488 
+.03521 

+04037 

+0.010  6 

4-OOO4.  8 

—  0047 
—jooy? 

2        O      2   —  2 

+.000  02 

—  I                2  —2 

+.000028 

+.000  19 

+  ooo  52 

O               2—2 
I               2  —2 

+.0000478 
—  .000002 

+.000  20 

—.000262 
—  ooo  04 

—.OOOIlS 

—.0000042 

—.00031 

—  0.000  169 

+XXXM32 

—I   —        2  —2 

—  .000400 

—  .00044 

O  —        2  —2 
I   —        2  —2 

—.000  4506 
+.000013 

—.001  55 

+.001  256 
-j-  ooo  24 

+.000337 

+  .000  7327 

+.00225 

+0.000  504 

+o  00026 

—.000406 

O       O     O        2 

—  .000  10 

.OOO  17 

+  00008 

.OO45 

FUNCTIONS  OF  LUNAR  COORDINATES. 

TABLE  XXXIX. 
FUNCTIONS  OF  COORDINATES  REFERRED  TO  MEAN  MOON. 


in 


Arg. 

*,' 
Brown. 

Brown. 

Brown. 

2^i!?i 
Brown. 

7~\fl;  2      w  2\ 
.Z-/1  V.    ~~Yl   ) 

Delaunay. 

2-Z?(Cl!0I) 
Delaunay. 

f   g'  *  V 

cos 

cos 

cos 

sin 

cos 

sin 

O       O     O       0 

i      o    o     o 

2       O     O       O 

—  i      i    o     o 

O        I     O       O 
I        I     O       O 
—  1020 
O       O     2       O 
I        O     2        O 
—  2        O     2   —  2 
—I        O     2   —  2 
O        02—2 
I        O      2   —  2 
2        O     2   —  2 
—  I         12—2 
0        12—2 
I         12—2 
—  I    —  I      2   —  2 
O   —  I      2   —  2 
I    —I      2—2 
I        O     O        2 
0        O     O        2 
—  1002 

—2      04—4 
—i      04—4 
004  —  4 

+.992  529 
—.10850 
+.004504 
—.000  35 
+.000  217 
+.000  19 
—.000  ii 
+.003  97 

.OOOOOO 

—.000  637 
—.01990 

—.012  13 

+.00069 
—  .00002 
+.00006 
+.000092 

.OOOOO 

—  .00080 
—.000875 

.OOOOO 

—.00005 
—.00021 

.OOOOO 

+.000  187 
+.00006 

+.OOOO2 

+.006299 
+.00033 
—.005989 
—.00035 
+.000049 

+.00035 
—.00022 

.OOOOO 
+.OOO22 
+.002444 
+.OOI  O2 
—.00244 

—.001  15 

—  .OOOO2 

+.986230 
—.10883 
+.010493 

.OOOOOO 

+.000168 

—  .000  16 

+.000  II 

+•00397 
—.00022 
—  .003081 
—  .02092 
—.00969 
+.00184 

.OOOOO 

+.00006 
+.000036 

—  .OOOOO 

—  .00080 
—.000721 
.00000 
—.00005 
—.00019 
+.00003 
+.000440 
+.00029 
+.00008 

-.007345 
+.00195 
—.00018 
—.001  36 
+.000198 

+.00054 
—  .00021 
—.000  ii 
+.00003 
—.00670 
—.04209 
—.03863 
+.00632 

+.00012 
—.00091 
+.000587 
—.OOOI2 

—  .001  50 
—.002889 
+.00055 
—  .000  14 
—.00024 
+.00006 
—.000  ii 
+.00264 
+•00074 

+•217  74 
—.004501 
—.OOI  IO 
-.006386 
—.00070 
—  .OOO  12 

—.003  95 

+.OOO  22 

+.001  843 
+.04434 
+.018  19 
—.00082 
+.00004 
—.00024 
—.000154 

.OOOOO 

+.001  97 
+.001336 

.OOOOO 

—.00005 
—.00056 

.OOOOO 

—.000  175 
—.00016 
—  .00006 

—  .00070 

+.OOO  12 

—.00384 
—.005  552 
—.002  10 
+.00041 
+.00016 
—.00003 
+.00432 
+•09472 
+.065  56 
—.00303 
+.00019 

+.OOI  12 

—.000  553 
+.00007 
+.00441 
+.005  097 
—.00026 
—  .000  1  1 
—  .00060 
+.00008 
—.00102 
—  .00051 
+.00004 

+  .O00056 
.OOOOOO 
.OOOOOO 

—  .000154 

.00000 
—JOOO  O2 

—.00003 
—.coo  253 
—  .00023 
—.00006 

Arg. 

a  A1 

d.tf 

**J& 

dj£ 

d.tf 

23  A?,  !   5-cs 

d-c 

de 

de 

de 

dr 

dr 

dr 

de 

dr 

g  g'   X  X' 

COS 

cos 

sin 

COS 

COS 

sin 

cos 

cos 

o     o    o     o 

I        O     O       O 
2000 
—  I        I      O       O 
O        I      O        O 
I        I     O       0 
—  I        O     2        0 
0        0     2        O 
I        O     2        0 
—  2        O     2   —  2 
—  I        O     2   —  2 
0        02—2 
I        02—2 
O        I      2    —2 
—  I    —  I      2   —  2 
O   —  I      2—2 
O        0      0        2 

—0.058  598 
—  1.9784 
+O.I63  69 
—  O.OO64 
—  0.000419 

+0.001  8 

+.227  700 

+.0084 

—•21735 

—.178059 

+.01  1  7 
+.0006 

—.000  281 

—  .0247 

—.000695 
—  .0076 

+.179588 

—  .0192 

+3.040  o 
—0.16464 
—0.023  5 
—0.003  478 
—0.015  9 

—0.002  0 

+000040 

+.001  537 

—.000067 

—.000014 

+.001786 

+.000004 

—.0039 

+.0107 

—.0054 
—.1948 
+.016  I 

+.01  1  5 
+.0018 
—  -0039 

+.0290 

—.1781 
—.0098 

+.195  6 

+.0039 
+.08881 
+.017  7 
—.087766 
—.021  7 
+.000877 

+0.005  8 
+0.0263 
+0.7969 
—0.066  474 
—0.0087 
+0.000  230 
—0.036  2 

—  O.OO2  870 

—.010  7 

—  0.023  02 
—0.360  o 
+0.067  295 
+0.014  3 
—  o.ooo  620 
—0.014  7 
+0.003  395 

+.0053 
+.014  604 

—.013  7 
—  .016  720 

+.000597 

+.000  i 

—.013  2 

—.OOO  121 

—.000044 

+.000  106 

—.004482 

+.000375 
—.0090 

+.000049 

+.000432 
—  XG44 

+•0132 

112 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


TABLE  XL. 
FUNCTIONS  OF  COORDINATES  REFERRED  TO  MEAN  SUN. 


Arg. 

P-lf 

/>'-3C2 

2651 

D(?-f) 

£>(/>2-3C2) 

2Z>,-jy 

g    g'  *     * 

COS 

cos 

sin 

cos 

cos 

sin 

o     o   o     o 

I       O     O       O 
2000 
—  I        I     0       0 
O        I     O       O 
I        I     O       O 
—  I        O     2        O 
O       O     2        O 
I        O     2        O 
—2       O     2  —2 
—  I        O     2  —2 
O       O     2  —2 
I        O     2  —2 
2        O     2  —  2 
—112  —2 
O        I      2   —2 
112   —2 
—  I   —I     2  —2 
O   —  I      2   —  2 
I    —I      2—2 
I        O     O       2 
O       O     O       2 
I        O     O  —  2 
O       2     O       O 

—  .01396 
—.031  26 
—.00250 
+.00013 
—.000933 
-.00135 
—.00004 
—.00038 
—.00002 
+.007  809 

—.163  02 
+.086I9 

+.05444 
+.00300 
—.00055 
—.003  18 
—.00042 
+.00027 

+.OO3  22 

+.00054 
—.00020 
+.00396 

+.OOO  IO 

—.000  059 

+.99074 
-.10860 

+.00096 
—.00071 
+.000264 

+.00055 
—.00032 
+.01196 
+.00066 
+.001807 
—.018  60 
—•01399 
—.00042 
—.00002 
+.00015 
+.00003 
.00000 
—00087 
—.00006 
.00000 
—.00032 
—.00084 
—.00004 
+.000005 

—.033  92 
—  .00243 
—  .00008 
—.001  123 
—  .00134 

+  .000  10 

—.00038 
—  .00002 
+.007  185 
—•163  54 
+.98611 

+.05444 
+.00300 
—.00056 
—.003  17 
—.00042 
+.00028 
+.003  14 
+.00054 
+.00024 
—.00396 

+.OOO  IO 

—.000059 

—.052095 

—.063  73 
—.005  55 
—.00058 
—.003  423 
—.003  04 
—  .00001 
—  .00042 

—  .OOO  12 
+.00030 
+.00291 
—.00700 
+.OOO62 
—.00003 
—.00260 
—  .00276 

—.00077 
+.001  33 
+.00289 
+.001  24 
+.00003 
—.000  13 
—.00031 
—.000  258 

—.004447 
+.004  62 
+.00007 
—.001  76 
+.000  710 
+.00126 
—  .00009 
—.000  15 
+.000  01 
+.003  73 
—.035  03 
—.04784 
—.001  53 
—  .00007 
—aoo  47 
+.00089 

+.OOO  01 

—  .001  40 
—.00398 
—.00015 
—.00076 
—.001  52 
+.00008 
+.000036 

-.07308 
—.005  47 
+.001  37 
—.004  563 
—.00289 
—  .00001 
—.00042 

—.OOO  12 
—  .00060 
—.00025 
—.O07  70 
+.00062 
—.00003 
—.00260 
—.00276 

—00077 
+.001  33 
+.00289 
+.001  23 
—.00003 
+.00013 
—.00031 
—.000266 

Arir 

a(«-/,') 

^-ac2) 

#3 

a^-tf) 

3(/>2-3C2) 

aej 

Arg. 

de 

de 

2  de 

dr 

^ 

df 

£•    g1  *    * 

cos 

cos 

sin 

cos 

cos 

sin 

_..  ct7  r^r 

I        O     O       O 

-0.565  7 

—1-955  i 

-0.6088 

+.0094 

+.050  2 

+.0095 

O        I     O       O 

+0.004  519 

+0.001  109 

+O.OO6  22O 

+.000  287 

—.000  161 

+.000469 

058  o 

o  003  6 

—  OI71 

-4-.cei6 

—  .CIS  I 

_j_  QIQ  6 

—  I        O     2  —2 
O       O     2  —  2 

—2.964  i 
—0.2863 

—0.343  1 

—0.021  5 

—2.964  2 
—0.2864 
if)  rfjfi  Q 

+.0182 
—.1782 
0066 

+.0052 
+  .0410 

+/>l83 
—.1780 

—  006  4 

-f-OOO88 

—  TOt  2 

PART  IV. 

DERIVATION   OF  RESULTS. 


CHAPTER   VII. 

CONSTANT   AND   SECULAR  TERMS. 

We  recall  the  arrangement  of  the  present  work.  In  Part  I,  the  general  equations 
have  been  formed,  the  theory  outlined,  and  the  methods  developed  so  far  as  could 
be  done.  Nearly  all  the  fundamental  quantities  were  developed  as  sums  of  prod- 
ucts of  two  factors,  one  factor  of  each  pair  being  a  function  of  the  Moon's  coordi- 
nates, the  other  a  function  of  the  coordinates  of  the  planets.  The  latter  functions 
are  developed  in  detail  in  Part  II,  one  chapter  of  which  is  devoted  to  the  develop- 
ments of  the  coefficients  of  the  direct  action,  the  other  to  the  coefficients  of  the 
indirect  action.  In  Part  III,  Chapter  VI,  have  been  developed  the  numerical 
functions  for  the  lunar  coefficients.  These  are  the  same  for  both  actions.  The 
present  concluding  Part  is  devoted  to  the  combination  of  these  factors  and  the 
derivation  and  discussion  of  results. 

We  may  divide  the  matter  of  this  part  into  three  chapters.  In  the  first  chapter 
we  consider  the  terms  not  purely  periodic.  By  a  purely  periodic  term  is  meant 
one  of  which  the  coefficient  of  the  sine  or  cosine  is  constant.  We  may,  therefore, 
define  the  terms  to  be  first  considered  as  constant  and  secular,  two  classes  which 
need  not  be  considered  separately. 

§  54.  The  arguments  on  which  the  planetary  and  lunar  factors  depend  are  all 
distinct  except  g',  which  is  common  to  both.  It  follows  that  no  constant  or  secular 
term  in  the  variations  of  the  elements  can  arise  by  the  multiplication  of  factors 
depending  on  any  other  variable  argument  than^''.  In  all  cases  in  which  another 
argument  than  this  enters  into  either  factor,  the  results  will  be  periodic  in  form,  the 
coefficient,  however,  having,  in  the  general  case,  a  secular  variation.  Since  no 
terms  of  the  class  in  question  contain  /,  TT,  or  0,  they  give 

D.  =  o  D    =  o  Z>7  =  o 


To  form  the  constant  and  secular  terms  we  begin  by  collecting  those  planetary 
factors  which  are  either  constant  or  depend  on  the  argument  g'.  We  shall  con- 
sider the  direct  and  indirect  actions  separately.  The  planetary  factors  for  the 
direct  action,  as  collected  from  §  42,  with  some  revision  of  the  numbers  there 
found,  are  shown  on  the  next  page. 

"5 


Il6  ACTION   OF  THE   PLANETS  ON  THE   MOON. 

FACTORS  FOR  DIRECT  ACTION. 

Action  of  Venus. 

'  =  +  5".  9045  +  o".44  cosg-'  —  o".n  sing-' 
=  —  3  .4072  —  0  .30cosg-'+o  .07  sing-' 
==  +  o''.33  sing-'  +  o".O3  cosg-' 

Action  of  Mars. 

=  +  o".0468  —  o".O2O  cosg-'  —  o".O24  sing-' 
=  —  o  .1006  +  0  .028  cosg-'  +  o  .029  sing-'  (18) 

=  +  o".oiosing-'  —  o".oo8  cosg-' 

Action  of  Jupiter. 

\o*MK=  +  o".O9i3  —  o".oo2  cosg-'  —  o".o32  sing-' 
%io3AfC——  2   .1348  +  0  .oO4Cosg-'  +  o  .062  sing-' 
=  +  o".oo5  sing-'  —  o".O3O  cos  g1 

Action  of  Saturn, 
=  +  p".ooi3  £ioWC'  =  —  o".i040  io*J/Z>  =  o 

The  corresponding  factors  for  the  indirect  action  have  been  combined  for  the 
five  disturbing  planets,  Venus  to  Uranus.  From  the  combined  values  of  G,  J, 
and  7,  reached  in  §  44,  we  find,  including  Uranus,  but  omitting  Mercury: 

io*m*G  =  +  o".459  +  o".36  cosg-'  +  o".o6  sing-' 
ios/w2/=  +  o  .153  +  0  .12  cosg-'  +  o  .02  sing-' 
io'wz'/=  —  o  .03  sing-' 

§  55.  Lunar  Factors.  If,  for  brevity,  we  put  F  for  any  one  of  the  three  lunar 
factors,  say 

F=?-f  F'={t-tf*  F"  =  2fr  (19) 

the  terms  of  the  fundamental  equations  (42)  or  (57)  corresponding  to  each  F  will  be  : 


From  the  tabular  values  of  the  functions  of  the  coordinates  and  their  derivatives 
in  Table  XL,  p.  112,  noting  that  symbolically,  D'  —D-\-2,  we  have  the  following 
values  of  the  terms  of  these  functions  which  are  independent  of  the  lunar  arguments 

F=  £2  —  ??2  =  —  .013  96  —  .000  933  cosg-'  —  .000  06  cos  2g-' 
F'  =  p*  —  3?2  =  .990  74  +  .000  264  cosg-' 
F"  —  2^rj  =  —  .001  123  sing-'  —  .000  06  sin  2g-' 


CONSTANT  AND  SECULAR  TERMS.  117 

D'F  =  D'(?  —  r)2)  =  —  .080  02  —  .005  288  cosg'  -  .000  38  cos  2g' 
D' F'  =  -D'(p2  —  3?3)  =  1-9770  +  .001  239  cosg-'  +  .000  04  cos  2g' 
D' F"  =  iD'ty)  =  —  .006  809  sing'  —  .000  39  sin  2g' 


de  '  de 


=  +  .1705  +  .001  109 


dF" 

-^  —  =  —=  —  =  +  .006  220  sin  g' 
de         de 


=  +  .0154  +  .000  287  cos  g' 


<5375  ~  '0°°        C°S 


=  +  .000469810^-' 

The  factors  a,,  e,-,  etc.,  are  derived  in  §  14,  and  found  in  (26).    From  the  preceding 
scheme  we  find  by  using  the  preceding  values  and  their  derivatives  in  (20) 

FI  =  —  0.1641  —  .010  92  cosg-'  —  .000  77  cos  2g' 

Ft=  —  2.1194  —  .086  40  COS£-'  +  .OOO  OI   COS  2g' 

FS  =  —  0.0859  —  -001  63  cos  g' 
FI  =  +  4.0086  +  .002  49  cosg-'  +  .000  08  cos  2g' 
FJ  =  -3-  3J42  -.021  36  cos  g'  (20) 

FJ  =  +  3.0093  —  .000  90  cosg"' 

F"  =  —  .013  89  sin  g'  —  .000  79  sin  2g' 

FJ'  =  —  .118  94  sing-'  +  .000  01  sin  2g' 

F"  =  —  .002  64  sing-' 

§  56.  Secular  motions  of  I,  TT,  and  6.     The  function  Aflf,as  defined  in  §  20,  may 
now  be  written 

MH=  MKF-  \MCf  +  MD.F" 


and  introducing  the  linear  functions  of  its  derivatives  which  we  have  just  formed 
we  have  from  (42) 

DJ,  =  -  MKF,  +  IMC^F'  -  MDF>' 

Djf^  =  -  MKFi  +  \MCFJ  -  MDF?  (21) 

DJ0  =  -  MKF,  +  \MCFj  -  MDFl' 

of  all  which  factors   we  have  just  given  the  numerical  values.     For  the  indirect 
action  the  second  members  are 

..-(1=1,2,3)  (22) 


Il8  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

Performing  the  multiplications  we  find  the  following  secular  motions  of  /„,  TTO,  and 
00  arising  from  the  terms  of  direct  and  indirect  action  under  consideration. 


Direct  Action  of  io5/?n(/o 

Venus  —  12.66  +  23.846  —    9-747 

Mars  —    0.41  +    0.433  —    0.300 

Jupiter  —    8.54  -}-    7.269  —    6.416 

Saturn  —    0.42  +    0.346  —    0.318 

Uranus  —    o.oi  -f-    0.007  —    0.006 

Sum  —  22.04  +  31-9°I  —  16.786 

Indirect  +    0.54  —    J-495  4-    0.422 

Total  —  21.50  +  30.406  —  16.364 

Taking  the  Julian  century  as  unit  of  time  n  =  8400.     The  centennial  motions 
arising  from  the  factors  here  employed  are  therefore: 

Centennial  motion  of  /„  =  --  i8o".6,  of  TTO  =  +  255.41,  of  00  =  —  137.46.       (23) 
From  the  vanishing  of  Dta,  Dte,  and  Dty  we  have 

Sti  =  const.  STTJ  =  const.  B0l  =  const. 

the  constants  being  functions  of  the  arbitrary  constants  of  integration,  determined 
at  the  end  of  this  chapter. 

§  57.  Terms  arising  from  the  secular  variation  of  the  earth's  eccentricity, 
Both  the  direct  and  indirect  actions  contain,  in  rigor,  terms  of  this  class.  They 
enter  into  the  direct  action  because  the  direct  action  of  the  planet  on  the  Moon  varies 
with  the  variation  of  the  orbit  of  the  earth  around  the  Sun.  But  the  effect  of  this 
variation  is  found  to  be  so  slight  that  it  will  be  left  out  of  consideration  in  the 
present  work.  We  therefore  begin  with  the  indirect  action.  The  terms  of  the 
coefficients  G,  /,  and  J,  on  which  the  action  depends,  have  been  developed  in 
Chapter  V,  §  44. 

Our  fundamental  quantity  for  the  indirect  action  is  H'  of  §  25,  of  which  the 
only  terms  required  are 


H'  =  -  G(?  -  O  -J(P*  -  3O  +  2/f,  =  -  GF-JF'  +  IF"  (24) 

The  terms  of   G,  /,  and  /  required  for  the  present  purpose  are 

G  =  G£e'  I  =  7,Ac'  /  =  J^e' 

Glt  /i,  andyi  being  found  in  §  44  and 

be'  =  -  8".595  T—  o".o26oT2  =  -  8".  595  T(i  +  .00302  T) 
The  secular  terms  of  these  coefficients  thus  become 


+  i".8^  cos  2g-')T(i  +  .00302  T) 
/=(4-o".io8o4-6".44cos£-'  +o".32cos  2g')T(i  +  .00302  T)  (25) 

/=  (-  25".45  sin^-'  -  i".84  sin  2g')T(i  +  .00302  T) 


CONSTANT  AND  SECULAR  TERMS.  119 

Using  these  values  in  (56)  we  find 


If  we  put  G',J',  and  /'  ior  the  coefficients  of  Tin  (25)  we  shall  have  from  (20) 
and  (22)  the  following  computation  for  the  secular  accelerations  from  the  funda- 
mental equations  (57),  in  which  only  the  non-periodic  terms  are  to  be  used  : 

[/]  =  G'F,  +  J'F{  -  I'F,"  =  +  o".2467 
M  =  G'F,  +  J'Fj  -  I'F,"  =  -  i  .6378 
Iff]  =  G'F,  +  /'/7  -  f'F,"  =  +  o  .3246 


Then,  postponing  the  terms  in  Z12 

DJ=m*[l]T  Djr=m*\*]T  DJ  =  m2[0]T  (26) 

The  terms  in  T2  in  (25)  are  only  those  arising  from  the  term  of  e'  in  T2.  To  find 
the  complete  values  we  note  that  all  the  terms  of  [/],  [TT],  and  [0]  contain  e'  as  a 
factor,  and  may  therefore  be  expressed  in  the  form  e'k,  k  being  a  quantity  which, 
though  containing  minute  terms  in  e'2,  may  be  regarded  as  a  constant.  Then 

D  e' 
/?,[/]  =  kDff  =  [/]  -^-  =  -  .002495  [/] 

and  the  actual  values  of  [/],  [TT],  and  [0]  may  be  written  in  the  form 

IX]  =    [/]„(!—  .00250  T)  (I   +.00302  T)=    [/]„(!  +   .  0005  2  T) 

[/„],  etc.,  being  the  values  computed  above.  Multiplying  by  T  we  find  that  the 
terms  of  Dtl,  D,TT,  and  DtO  in  T2  are  found  from  those  in  T  by  multiplying  the 
latter  by  the  factor  -|-  .00052  T. 

Taking  the  Julian  century  as  the  unit  of  time,  w2«  =  46.998,  whence 

Dl/=  +  ii".6oT  +  o".oo6oT2    Dtir=  -76".  98  T-o".  040  T1    Z>,0=  +  i5.  25^+0.00797^ 
Then  by  integration 
8/=5".8or2  +  o".oo2or3    S7r=-38".49712-o".oi3r3    W  =  f  .62  T*  +  o"  .0026  T*  (27) 

This  value  of  the  secular  acceleration  of  the  mean  longitude  is,  I  believe, 
markedly  smaller  than  any  heretofore  found.  Delaunay's  last  result  was  6".n, 
which,  reduced  to  the  now  adopted  value  of  the  secular  diminution  of  e',  would 
become  6".O2.  The  necessity  of  using  Delaunay's  development  of  the  parallax  in 
forming  the  Z>'s  of  some  of  the  coefficients  leads  to  some  uncertainty  in  the  present 
result.  But  my  rough  estimate  would  lead  to  the  conclusion  that  the  uncertainty 
should  be  less  than  one  per  cent,  of  the  whole  amount.  The  question  of  the  pre- 
cision of  the  value  here  reached  I  must  leave  to  other  investigators.* 

*  As  this  work  is  going  through  the  press  the  author  notices  that  Brown's  value  found  in  Monthly  Notices  Royal 
Astronomical  Society,  vol.  LVir,  is  reduced  from  $".<)i  to  5".8i  when  the  now  adopted  Die'  is  used. 


120  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

§  58.  We  have  next  to  consider  the  secular  variations  of  the  periodic  terms  in 
general.     Taking  any  set  of  such  terms  depending  on  any  argument  N 

?*  —  i;*=  2p  cos  JV  p*  —  3%*  =  2q  cos  JV  2%i)  =  K4  sin  JV 

we  shall  have  the  terms  of  H'  in  (24) 


(i".o72/-o".2i6?)  Tcos  A'-(i9".3/+6".4sr+  12".7«4)  Tcos  (W-g1 


Forming  the  partial  derivatives  of  these  terms  of  H'  as  to  /,  TT,  0,  a,  e,  and  y,  and 
carrying  them  into  the  fundamental  equations  (64)  and  (65)  by  the  processes  of 
§  §  22  and  23  we  shall  be  led  to 

D  ,a=m\—i".o'j2ap+o".2i6ag)Tsin  7T+  w2(I9"-3a/+6"-4a?+  I2"-7a*<)^sin  (W—g1) 

(28) 


with  similar  equations  for  e  and  y  formed  by  writing  e  and  g  respectively  for  a. 
Also,  we  shall  have 


(20) 
-7«l(i9".3Z'  +  6".4Z"-  i.2".>]Lt)  '    v 


with  similar  equations  for  D^  ir0  and  D,u  6Q  formed  by  writing  P  and  R  respectively 
forZ 

§  59.  The  special  values  of  N  of  most  importance  in  the  present  connection  are 


on  which  depend,  respectively,  the  constant  term,  the  annual  equation,  the  equation 
of  the  center,  and  the  evection 

CASE  I;  7V=o. 

The  factors  for  Z^a,  Dnte,  and  Dnty  all  vanish.  The  values  of  the  Z-coefficients 
are  found  in  the  first  line  of  Table  XLIX,  p.  147.  The  first  or  purely  secular  term 
of  (29)  has  already  been  computed.  The  remaining  terms  give 

Z>/0=  ws«(38".6Z'  +  i2".9L")Tcosg-' 
Z?(7r0  =  »22«(38  .6P'  +12  .<)P")Tcosg' 
Dfa  =  m2n  (38  .6ft'  +  1  2  .9^?")  T  cos  g' 

Substituting  the  numerical  values  of  Z,  /"*,  /?,  and  7«2«  =  47.oo; 
/?,/„=  +  1062"  T  cos  g'  Z>(TTO  --  296i"Tcosg'  £>&=  +82f'Tcosg-' 

We  cite,  for  convenient  reference,  the  following  indefinite  integrals 

/it  C  if 

t  sin  titdt  =  —  ,  sin  N/  --  cos  N£  I  /  cos  wtdt  =  —t  cos  N/  +  -  sin  N/ 

N  N  J  N  N 


CONSTANT  AND  SECULAR  TERMS.  121 

The  unit  of  t  in  these  equations  being   100  years,  N   is  the   motion  of  g'  in  this 
period,  for  which  we  may  take  200  TT,  or  N  =628. 
Integration  by  the  above  formulas  then  gives 

S/0  =  +  i".  69  T  sin  g'  +  o".oo3  cos^' 

87r0=—  4  .71  Ts'mg-'  —  .ooScosg-'  (30) 

S00=  -f  i   .32  T  sing'  +  .002 


CASE  II;  N=g';  the  annual  term. 

Here  also  the  variations  of  a,  e,  and  y  vanish,  so  that  only  those  of  /0,  ir0,  and 
00  are  affected.  Carrying  into  the  equations  (29)  the  numerical  values  of  the  lunar 
coefficients  for  the  Arg.  g'  we  find,  dropping  the  constant  terms,  which  have  been 
already  computed, 


m^n  T(o".oo6i  cosg-'  —  o".o82  cos  2g')  =  o".2gTcosg'  —  3".8o  Tcos  2g' 
m2nT(o  .0485  cos  g'  +  o  .131  cos  2g')  =  2  .28  T  cos  g'  +  3  .^Tcos  2g' 
nfn  T(o".ooio  cosg-'  +  o".O22  cos  2g)  =  o".tf  Tcosg'  +  i  .03  Tcos  2g' 

Then,  integrating,  and  dropping  insignificant  constant  coefficients 

8/0  =  -f  o".ooo47  Tsing'  —  o".  0036  T  sin  2g' 
8-7r0  =  +  0.0037  Tsin  g-'  —  0.0028  7"sin  2g' 
S0g=  +  0.0008  T  sin  g-'  —  o.  0016  T  sin  2g' 

CASE  III;  N-ff. 

For  this  argument  I  have  used  the  following  preliminary  values  of  the  lunar 

coefficients,  differing  from  those  of  Tables  XLVIII  and  XLIX  by  amounts  here 
unimportant 

L'  =  —  0.1197              L"  =  —  0.1962  Z.4  =  —  0.2648 

P'=+  5.403                P"  =  +  18.734  PI  =  +  "-688 

7?'==  —  0.0028              R"  =  —  0.1431  7?4  =  —  0.0560 

ap  =—0.03223             &q  =  —  0.11049  a*4=—  0.06970 

e/  =  —  0.300  45            6^=  —  1.03340  e*4  =  —  0.650  66 

gp  =  +  o.ooo  09            g?=  +  o.ooo  31  g*4=  +  o.ooo  10 

Carrying  these  values  into  the  equations  (28)  and  (29)  we  find,  for  the  terms  depend- 
ing on  the  argument  g  alone, 


Dte=  + 
,  =  +  ^".o^Tcosg-  Z?,7r0  =  -  82".  2  Tcosg- 


For  the  motion  of^f,  N  =8329. 
Integration  then  gives 

8ot=  —  o".ooo  062  T  cos  £-+(7  4"-r-  io'°)sin^-  Be  =  —  o".oo$6Tcosg- 

S/9  =  +  o".ooo  485  Ts'm  g-  b*0=-  o".oo9  87  Tsin  g- 


122  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

We  drop  the  terms  with  constant  coefficients,  owing  to  their  minuteness,  and 
find,  with  #  =  8400; 

Sn  =  -  |«Sa  =  +  o"-78  T  cosg 
Then  by  integration 

=  +  o".ooo 


This,  added  to  8/0,  gives  for  the  entire  term  in  S/ 

8t=  +  o".ooo  578:Tsin£-  (31) 


§  60.  In  order  to  determine  the  complete  expressions  for  the  coordinates  them- 
selves, the  terms  computed  in  the  present  section,  together  with  those  which  may 
be  found  in  a  similar  way  for  the  other  periodic  terms,  are  to  be  carried  into  the 
expression  for  the  Moon's  true  longitude  in  terms  of  the  elements.  I  have  not, 
however,  deemed  it  necessary  to  do  this  in  the  case  of  the  secular  variations  of 
the  periodic  terms,  because  these  can  be  most  readily  determined  by  varying  the 
value  of  e'  in  the  Delaunay  or  Brown  expressions  for  the  Moon's  longitude. 

I  have,  however,  computed  the  preceding  variations  of  some  terms  owing  to  the 
theoretical  interest  which  attaches  to  the  relations  implied  by  the  equality  of  the 
result  of  the  present  method  to  those  of  the  other  method.  The  two  methods 
correspond  to  the  two  methods  by  which  the  secular  acceleration  ma}'  be  deter- 
mined. In  Action,  p.  191,  it  is  shown  that  the  secular  acceleration  of  /,  TT,  and  6 
may  be  derived  from  the  secular  change  of  e'  by  determining  the  corresponding 
secular  changes  in  a,  e,  and  y.  This  theorem  has  been  discussed  and  extended  by 
Brown  in  his  paper  on  Transmitted  Motions  and  Indirect  Perturbations.* 

By  this  method  the  secular  variations  in  question  appear  as  variations  of  »,  rr^  and 
&i,  the  latter  being  functions  of  the  variables  a,  e,  and  y.  But,  in  the  present  theory, 
a,  e,  and  y  remain  constant  so  far  as  the  secular  change  of  e'  is  concerned,  and  the 
changes  are  thrown  wholly  upon  /0,  TTO,  and  60. 

There  is  therefore  a  seeming  contradiction  in  that  the  lunar  elements  a,  e,  and  y 
are  affected  by  a  secular  variation  in  one  theory,  while  in  the  other  they  are  prac- 
tically constant.  Referring  to  Brown's  paper  for  the  theory  of  the  subject  it  will  be 
instructive  to  show  the  relation  between  the  two  methods. 

In  what  I  have,  for  brevity,  called  the  Delaunay  solution  of  the  problem,  the 
Moon's  coordinates  appear  as  functions  of  the  lunar  elements,  introduced  as  arbi- 
trary constants,  and  of  the  Sun's  eccentricity,  which  is  regarded  as  a  quantity  given 
in  advance.  But,  when  the  action  of  the  planets  is  introduced,  the  solar  element 
e',  as  well  as  the  lunar  elements  a,  e,  and  y,  become  variable.  In  what  I  may  call 
method  A  of  treating  the  planetary  action,  which  was  that  adopted  in  Action, 
the  final  values  of  the  coordinates  as  affected  by  planetary  action  are  determined 
by  introducing  the  simultaneous  variations  of  all  four  elements  into  the  Delaunay 

*  Transactions  of  the  American  Mathematical  Society,  vol.  vi,  p.  332.  See  also,  Monthly  Notices,  Roval  Astro- 
nomical Society,  vol.  LVII. 


CONSTANT  AND  SECULAR  TERMS.  123 

expressions.  But  in  method  J3,  adopted  in  the  present  work,  the  entire  variations 
have  been  thrown  upon  the  lunar  elements,  the  solar  elements  being  regarded  as 
constant.  In  the  case  of  the  periodic  perturbations  this  course  is  practically  a 
necessity,  owing  to  the  extreme  complexity  introduced  into  the  formulae  if  we  sup- 
pose the  coordinates  expressed  in  terms  of  the  value  of  e'  affected  by  periodic 
inequalities.  But  it  is  different  in  the  case  of  the  secular  motion  of  e'.  Here  it 
is  more  logical  to  consider  that  at  any  epoch  the  action  of  the  Sun  is  computed 
with  the  actual  eccentricity  at  that  epoch,  and  so  to  use  method  A. 

Not  having  done  this  in  the  present  work,  but  having  regarded  the  value  of  e'  at 
the  epoch  1850  as  a  fundamental  constant,  the  values  of  G,  J,  and  /,  though  func- 
tions of  e',  and  therefore  variable,  have  appeared  in  the  theory  as  constants. 

In  the  present  investigation  the  author  has  not,  for  want  of  time,  investigated  the 
modifications  which  would  be  made  in  the  problem  if  these  coefficients  were  taken 
as  affected  by  their  secular  variations.  One  reason  for  refraining  from  this  course 
was  that  the  determination  of  the  secular  acceleration  from  the  equations  given  in 
Action,  page  191,  require  a  much  more  extended  development  of  the  canonical 
elements  in  terms  of  e'  than  it  was  practicable  to  undertake  in  the  present  paper. 
The  question  is  therefore  left  to  others,  reference  being  made  to  Brown's  paper  on 
the  variation  of  given  and  arbitrary  constants.* 

A  comparison  of  the  secular  variation  of  the  coefficient  of  sin  g'  with  that  found 
by  Delaunay's  value  of  this  term  will,  however,  be  of  interest.  With  the  eccen- 
tricity of  1850  the  coefficient  of  this  annual  term  is  —670".  It  contains  e'  as  a 
factor,  the  portion  arising  from  higher  powers  of  this  element  being  unimportant  in 
the  present  case.  It  follows  that  the  secular  variation  of  the  coefficient  of  sin  g' 
in  8v  is 

-670"^  =  +  i".6>jT 


e' 


The  term  found  in  (30)  for  8/  is  i".69  T.     I  have  not  computed  8v  itself. 

The  two  methods  of  treating  the  effect  of  the  motion  of  the  ecliptic  are  related 
to  each  other  in  the  same  way  as  this  just  discussed.  Had  the  method  of  the 
present  paper  been  strictly  followed  throughout,  the  coordinates  of  the  Moon  would 
have  been  referred  to  a  fixed  ecliptic,  because  the  ecliptic  remains  fixed  when 
planetary  action  is  omitted.  But  it  was  seen  that  by  a  very  slight  and  easily  deter- 
mined change,  the  coordinates  could  be  referred  to  the  actually  moving  ecliptic,  and 
and  the  work  was  carried  on  accordingly.  In  concluding  the  work,  it  is  a  matter 
of  regret  to  the  author  that  he  did  not  investigate  the  question  whether  the  Moon's 
coordinates  could  not,  on  the  same  principle,  be  expressed  in  terms  of  a  varying 
solar  eccentricity,  ab  initio,  thus  simplifying  the  problem  in  conception  at  least. 
Owing,  however,  to  the  theoretical  interest  attaching  to  the  relation  between  the 
two  methods,  the  effect  of  the  motion  of  the  ecliptic  might  be  treated  by  both  methods. 

*  L.  c.,  vol.  iv,  p.  333. 


124  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

§  61.  Adjustment  of  the  Arbitrary  Constants.  The  problem  before  us  may  be 
outlined  thus.  The  preliminary  solution  of  the  problem  of  three  bodies  leads  to 
expression  of  the  Moon's  coordinates  as  functions  of  six  arbitrary  constants,  through 
the  intermediary  of  three  other  functions  of  these  constants  /,  IT,  and  6.  The  solu- 
tion in  terms  of  the  six  elements  a,  e,  y,  /,  TT,  6  takes  the  form: 


the  functions  <jt  being  of  a  form  not  necessary  to  specify  at  present.  As  already 
mentioned,  n,  TT^  and  6l  are  functions  of  a  (or  «),  e,  and  y.  The  solution  of  our 
problem  is  now  completed  by  adding  to  the  expressions  for  the  Moon's  coordi- 
nates =  v,  r,  and  /8,  the  quantities 

dv  -         dv  -        dv  .         dv  „ 


with  similar  forms  for  r  and  /8,  which  we  need  not  write.     For  our  present  purpose 
it  will  be  necessary  and  sufficient  to  consider  the  following  terms  in  v,  the  true 

longitude. 

v  =  /  +  2e  sin  (/  —  TT) 

We  then  have 

dv      dv  dv 


dv  .  dv  dir  dv      dv  dl      dv  dir 

de~  °      dir  de  da~  dl  da      dir  da 

Substituting  in  (32)  and  emitting  unimportant  terms 

Bv  =  —  fyit(\  +  2e  cos£-)8a  —  2c  COS^TT  +  (i  +  ze  cos^)8/0  +  2  sing-Se 
We  put 

H'    8<*o'    ^o'    8/o' 

the  arbitrary  constants  to  be  added  to  the  perturbations  8a,  8e,  STT,  and  87.     We  then 
have  the  following  perturbations  depending  on  the  purely  lunar  arguments 

Sa  =  Sa0  —  o".ooi6  sing-  Be  =  Sc0  —  o".oi$o  cosg 

&I=  &tg  —  ",02i2nt  +  ".0059  sin£-  STT  =  STTO  +  ".o$i$nt  —  ".272 


Substituting  in  the  derivatives  we  have  the  result  that  the  mean  sidereal  motion 
of  the  Moon  is 

nt(i  —  o".O2i2  —  |S0a) 

We  now  determine  80a  by  the  condition  that  the  mean  motion  shall  be  repre- 
sented by  n.     Thus 

S0a  =  —  o".  0141  80«  =  o".o2i2«  =  +  178"  (33) 


CONSTANT  AND  SECULAR  TERMS.  125 

Also,  the  coefficient  of  sin  g-  in  the  expression  for  the  longitude  becomes 

2e  -f  o 


We  now  determine  S0e  by  the  condition  that  the  expression  for  the  coefficient  shall 
remain  unchanged.     This  gives 

V  =  —  o".oo3 

The  longitudes,  perigee,  and  node  being  given  by  the  equations 

ir  =  w,  +  V  e  =  ea  +  oj 

the  introduction  of  the  perturbations  of  the  elements  will  give  rise  to  the  increments 

»*,-£'&,  +  £>&  +  £'  *y  W,  =  f'  Bn  +  %  Be  +  %**, 

on  ffe  cy  dn  de  dy 

The  eftects  80e  and  80y  are  inappreciable.     Taking  only  80»  from  (33)  we  have 

STTI  —  —  .014  8oS0n  =  —  o".ooo  3i4«  &0t  =  —  .001  oiSan  =  —  ".ooo  O2i« 

Taking  the  century  as  the  unit,  the  adjustment  gives 

S7r1=-2".6  S0  =  -o".i8  and          Sir  --  2".6T         S6  --  o".i8T 


Adding  thereto  the  secular  terms  of  ITO  and  00  already  found,  we  have  the  following 
results,  tor  the  entire  secular  effect  of  the  action  of  the  planets  on  TT  and  6 

D,v  D& 

Direct  action  of  the  planets  Venus  to  Uranus  +  267".  97  —  141".  oo 

Indirect  action  of  the  planets  Venus  to  Uranus  —    12   .56  +      3  .54 

Total  action  of  Mercury  (/«  ==  io~7)  +      o  .45  —      o  .21 

Adjustment  of  elements  —      2  .64  —      o  .18 

Sum  +  253  .22  —  137   .85 

This  motion  of  the  perigee,  greater  by  5"  than  that  found  by  Brown,  goes  to 
confirm  his  conclusion  that  the  gravitation  of  the  Earth  does  not  deviate  from 
Newton's  law  of  the  inverse  square. 

§  62.  As  the  reason  for  the  last  correction  may  not  be  quite  clear,  it  may  be  of 
interest  to  state  in  a  general  way  how  it  enters  into  the  theory.  The  action  of  the 
planets  on  the  Moon  is  found  on  the  supposition  of  what  we  may  call  an  undisturbed 
orbit  of  the  Moon,  meaning  thereby  an  orbit  in  which  the  action  of  the  Sun  is  com- 
pletely taken  account  of,  on  the  supposition  that  no  other  extraneous  action  enters. 
We  thus  have  a  certain  mean  motion  n  determined  from  observations,  and  a  certain 
undisturbed  mean  distance,  a,  determined  by  the  relation  asn2  =  /i,  which  requires  a 
constant  A«  of  correction  to  the  mean  distance  computed  from  the  action  of  the 
Sun,  giving  rise  to  an  expression  for  the  constant  of  the  Moon's  radius  vector 
a  -)-  A^  =  #!  completely  representing  the  action  of  the  Sun  on  the  supposition  of 
no  planetary  action. 


126  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

It  is  with  this  mean  distance  a,  that  the  actions  of  the  planets,  both  direct  and 
indirect,  are  computed.  But,  as  a  matter  of  fact,  the  action  of  the  planet  modifies 
the  relation  between  a^  and  n,  so  that  we  must  change  either  the  mean  motion  or 
the  mean  distance  according  to  what  values  of  the  elements  we  assume.  If  we 
take  the  arbitrary  constants  so  that  the  mean  motion  remains  unchanged,  then  the 
actual  mean  distance  will  require  a  constant  correction  on  account  of  the  action  of 
the  planets.  If  we  regard  the  mean  distance  as  an  invariable  quantity,  then  there 
will  be  a  correction  to  the  mean  motion. 

It  follows  by  either  method  that  when  we  compute  the  motion  of  the  perigee  and 
node  under  the  action  of  the  Sun  alone,  we  must  make  one  or  the  other  of  these 
modifications  produced  by  the  action  of  the  planet,  and  determine  the  effect  upon 
the  motion  of  ir  and  0.  If  we  regard  the  actually  observed  mean  motion  as  that 
due  to  the  Sun  alone  then  we  must  introduce  a  correction  to  the  mean  distance,  and 
determine  its  effect  upon  -nv  and  6^  But  if,  which  is  the  more  natural  method,  we 
regard  the  mean  distance  of  the  Moon  as  the  given  actual  element,  then  we  must 
compute  that  part  of  the  motion  of  the  perigee  and  node  due  to  the  Sun  alone  with 
a  different  n  from  that  given  by  observation;  that  is,  with  a  value  found  by  sub- 
ducting the  planetary  effect  from  the  observed  value. 

We  ma)'  therefore  regard  the  corrections  --  2"  .64  and  —  o".i8  to  77^  and  0l  as 
reducing  ir^  and  0,  to  their  true  values  under  the  action  of  the  Sun  alone. 

§  63.  Secular  Variation  of  e.  If  we  require,  as  we  should,  that  the  coefficient 
of  sin  g  in  the  Moon's  true  longitude  should  be  represented  by  a  function  of  e  then 
the  expression  (31)  shows  that  this  element  will  be  affected  by  the  secular  variation 


This  being  less  than  o".oi  in  a  thousand  years,  is  of  no  practical  importance,  though 
of  theoretical  interest. 

It  may  also  be  remarked  in  the  present  connection  that  the  existence  of  this 
variation,  and  the  approximate  algebraic  expression  for  its  amount,  was  first  made 
known  by  Adams.* 

*  Monthly  Notices,  Royal  Astronomical  Society,  vol.  XIX,  p.  207. 


CHAPTER  VIII. 

SPECIAL  PERIODIC  INEQUALITIES. 

§  64.  Reduction  to  the  moving  ecliptic.  Since  when  the  Sun  is  the  disturbing 
body  the  plane  of  the  ecliptic  remains  fixed,  the  inequalities  of  the  coordinates  so 
lar  reached  are  referred  to  the  ecliptic  of  any  date  regarded  as  fixed.  The  only 
way  in  which  they  are  affected  by  the  motion  of  the  ecliptic  is  through  the  secular 
variations  of  the  coordinates  of  the  planet  arising  from  that  motion.  The  effects  of 
these  are  supposed  to  be  too  small  to  need  consideration  at  present.  It  is,  however, 
necessary  to  refer  the  elements  to  the  moving  ecliptic.  I  have  shown  in  §  4  how 
this  may  be  done  by  the  simple  device  of  adding  to  the  perturbative  function  the 
terms 

AT?  =  2z(pDixl  -  qDty^)  +  2  (qy  -  j>x)  Dft  (33) 


and  then  integrating  the  portions  of  the  differential  equations  thus  arising.  In  this 
expression  p  and  q  are  the  coefficients  expressing  the  speed  of  rotation  of  the 
ecliptic  around  the  axes  of  y  and  x  respectively,  and  are  found  by  putting 

II,  the  longitude  of  the  ascending  node  of  the  moving  on  the  fixed  ecliptic; 

K,  the  speed  of  rotation. 
Then 

p  =  K  sin  II  q  =  K  cos  II  (34) 

It  is  to  be  noted  that  K  is  here  used  as  the  speed  of  rotation,  and  not  as  the  actual 
angle  rotated  through.  It  is,  therefore,  of  dimension  Z1"1  and  the  expression  for 
AT?  is  of  dimensions  Z*2T^~~,  which,  by  introducing  the  dimensions  of  mass,  become 
identical  with  the  dimensions  of  P  as  hitherto  used. 

The  partial  derivatives  of  AT?  as  to  the  lunar  elements  are  to  be  taken  only  as  they 
enter  through  x,  y,  and  z,  so  that  the  Z>,  of  the  Moon's  coordinates,  the  latter  being 
called  for  this  purpose  xlt  yv,  and  zlf  are  to  be  regarded  as  numerically  given 
quantities. 

To  form  the  partial  derivatives  of  x,  y,  and  z  we  use  the  developments  of  these 
coordinates  in  terms  of  the  lunar  elements  already  given,  substituting  in  x,  y,  and  z 
the  values  of  £,  TJ,  and  £.  But  in  this  part  of  the  work  it  will  be  convenient  to  refer 
the  coordinates  x  and  y  to  a  general  fixed  X-axis,  instead  of  the  mean  Sun,  as  here- 
tofore. When  this  is  done  the  expressions  for  the  ratios  of  the  coordinates  to  a 
take  the  torm 

7,  =  2/&sinyV  £=2csinJV'  (35) 

127 


128  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

where  N  and  N'  are  of  the  general  form 

N=  il  +  />  +  i,0  +j'P  +/X          N1  =  i7/  -f  »,V  +  it'0  +//' 
the  indices  satisfying  the  conditions 

*  +  *,  +  *;  +  j  +  /,  =  o          ir  +  *,'  +  1/  +/  +  //  =  i 

Informing  the  Z>,'s  of  these  expressions  we  put  n,  n',  the  ratios  of  the  motion  of 
the  arguments  N  or  N'  to  n,  that  of  the  Moon.     We  then  have 


sin  TV  -^(Ji  =  anS.kn  cos  TV  -O^,  =  an"S.cn'  cos  TV' 

The  values  (34)  of  ^>  and  q  then  give 

//>,#,  -  gDtyl  =  —  anicZkn  cos  (TV  -  II)  (36) 

qy—px  =  aiCLk  sin  (TV—  II)  (37) 

Our  next  step  is  to  form  the  derivatives  of  z  and  qy—px  as  to  the  lunar  elements. 
The  partial  derivatives  as  to  z  are  found  from  the  last  equation  (35) 

Dz  =  alD'c  sin  TV'  ~=«2^sinTV'  f  =  «S^sinTV'  (38) 

cte  fo  dy  dy 

dz  dz  dz 

=-.  =  aZt'c  cos  TV'  a~  =  aSz'.V  cos  TV'  ~  =  a2»/c  cos  TV'  (39) 

Ol  CTT  VO 

By  differentiating  (37)  on  the  same  system  we  have 

D(qy  -PX)  =  aKLD'k  sin  (TV-  H) 

.,„  {^_n)  (4o) 


cos    y_ 


=  a«2^  cos  (TV-  H) 


We  next  have  to  form  the  products  of  (36)  by  the  derivatives  (38)  and  (39)  and 
of  Dtz±  by  (40)  and  (41),  and    form  their  several  sums.     We  thus  find  that  the 


MOTION  OF  THE  ECLIPTIC.  129 

combination  of  any  term  of  argument  TV7"  with  any  term  of  argument  N'  gives  rise 
to  the  following  terms  in  the  partial  derivatives  as  to  e  and  /: 


-^  «2w/c  I  nV^ nk^  \  sin  (N  -f  N' —  II)  +  02»/c  I  n'c^r  +  nk^  \  sin  (TV7"—  N'  —  II) 

cte  ,  oe  oe )  oe  Be } 

d~~^  =  a>nKck{m'  -  t'n}  {cos  (N+  N'-  H)  +  cos  (N -  N'-  II)}  (42) 

The  derivatives  as  to  log  a  and  y  are  formed  from  the  first  of  these  equations  by 
simple  substitution.  Those  as  to  ir  and  6  are  formed  from  the  last  equation  by 
writing  t\  and  /2  for  z,  and  z'^  and  z''2  for  i' . 

The  derivatives  thus  formed  being  substituted  in  the  fundamental  equations  the 
integration  of  the  latter  will  give  the  inequalities  of  the  elements.  It  will  be  con- 
venient to  use  the  following  formulae  of  substitution.  We  first  put,  in  the  combina- 
tion of  any  term  of  argument  JV  with  any  term  of  argument  JV' : 

ka  =  n'cD'k  +  nkD'c         k'  =  n'cD'k  -  nkD'c 


,   dk           Be  ,   dk           dc 

K  =  n'c  -^  —  (-  n«  ^-  k  '  =  n'c^  --  nk  =- 

oe           oe  de           oe 

*, 

,    dk         7dc  ,   dk         ,dc 

k  =  rf  c  jr-  +  nk  jr-  t'**n'c-s  --  n£^- 

oy            dy  ey            oy 


The  quantities  ka,  /&'„,  etc.,  will  then  be  the  coefficients  of  the  constant  factor 
crnK  in  the  expressions  for  the  derivatives  of  the  elements.  Substituting  AT?  for 
P1  in  the  differential  equations  (27),  p.  18,  the  latter  will  reduce  to  the  form 


Dp  =  (a/-,  +  OLJI,  +  a/,)*  {cos  (JV+  N'  -  II)  +  cos  (N-N1  - 

Df  =  (eft  +  e2k,  +  e3k,}  K  {  cos  (N  +JV'-U)  +  cos  (N-  N'  -  II)  }  (44) 

ft  =  (7,*,  +  7A  +  7s*f)«  {cos  (JV+  N'  -  H)  +  cos  (N-  N'  -  II)} 

sin  (^~  N>-  n)  +  («,*.'+  «?,*.'+  7,V)«  sin 
sm  (N-  JV'-U)  +  (of.'  +  e&+  t&')e  sin 

sin  (N-  N'  -  H)  +  (a/a'+  '&+  7^')*  sin  (^+  N>  ~  D) 

By  integrating  these  equations  we  shall  have,  in  the  case  of  each  argument,  a 
divisor  which  we  may  call  N,  equal  to  the  motion  of  the  argument  in  the  unit  of 
time.  The  quotient  K  -^-  N  expresses  the  angular  motion  of  the  ecliptic  during  the 
time  required  for  the  argument  to  move  through  the  unit  radian. 

In  the  above  differential  equations  we  substitute  for  a,,  et,  and  yh  their  numerical 
values  and  write,  for  brevity, 

Ct  =  2.023^  —  0.017^  —  0.0229^ 

C,  =  -  0.0301^  -  19.153*.  -  o.o2o£y  (45) 

Ct  =  0.0075^,,  +  o.oo26£e  —  5.570^ 


ACTION  OF  THE  PLANETS  ON  THE  MOON, 
with  similar  expressions  for  the  accented  quantities,  and 

Ca  =  2.023-6,  -  0.0301^  +  0.0075^ 

Ce  =  —  0.0168^,—  19.153^,  +  O.0026/6, 

Cv  =  —  0.0229/6,  —  0.0200/6,,  —  5.570/6, 


(46) 


We  also  put  for  brevity 

A  =  TV-  N'  - 


A'  =  TV+TV'  -n 


The  values  of  N  and  N',  the  coefficients  of  the  time  in  A  and  A'  respectively  take 
the  form 

N  =  (n  —  n')»  N'  =  (n  +  n')« 

and  the  differential  variations  become 


Dp.  =  Cjc  (cos  A  +  cos  A')         —  Z>,/0  =  C>  sin  A  +  C/K  sin  A' 


Df  =  Cjc  (cos  A  +  cos  A') 
Dty  =  Cyx(cos  A  +  cos  A') 

We  shall  then  have  by  integration 


K  1C 

:  =  -  Ca  sin  A  +  — ,  C0  sin 


if  if 

Se  =  -  Ce  sin  A  H — -,  Ct  sin  A* 

N  N 

57  =  ~  Cy  sin  ^  +  ~  Cy  sin  ^' 


—  Z>(TTO  =  C>  sin  .,4  +  C>  sin  ^4' 

—  Dt00  =  C>  sin  A  +  Ct'ic  sin  A' 


/,  =     C,  cos  ^4  +       C/  cos  A' 


K  K 

i  =  N      '  C°S        +  N"'       '  C°S 


><=-CtcosA  +  ^Qcos 


(47) 


(48) 


The  largest  terms  which  enter  into  the  theory  are  shown  in  Table  XLI«,  for 
Arg.  TV,  and  Table  XLI£  for  Arg.  TV'.  The  coefficients  of  the  principal  terms  of 
each  have  been  derived  from  the  numbers  given  in  Part  III. 

TABLE   XLIa. 
COEFFICIENTS  FOR  FORMING  px—qy ;  ARG.  TV. 


NA 

/      JT     0      /'      ff' 

k 

D'k 

dkldf 

dklBf 

nt 

Z        »!    '*     ^1      ^ 

VKjUf 

i 

I        O     O       O       O 

+  I.OOOOO 

+-995S 

+.9925 

—0.0588 

+•9955 

+•9955 

2 

2  —  I     O       O       O 

+  1.091  55 

+.0275 

+.0275 

+04980 

.0000 

+•0547 

3 

O        I     O       O       O 

+0.008  45 

—.0820 

—.0815 

—  14934 

.0000 

—.00069 

4 

—  I        2     O       O       O 

—0.983  i 

+.0004 

+.0004 

+0.0142 

xxwo 

—.0004 

5 

2  —  I     O  —  I        I 

+1.9168 

+.O002 

+.000  1 

+0.0038 

.0000 

+.0003 

6 

O        I     O        I    —I 

+0.083  2 

—.0004 

.0000 

—0.0070 

.0000 

.0000 

7 

—I        02        O       O 

—  1.0080 

+.OO2O 

+.OO2O 

+O.OOO2 

+.0980 

—  .0020 

8 

—  I        0020 

—0.850  4 

—.0083 

—.0081 

+0.0052 

+.0072 

+•0071 

MOTION  OF  THE  ECLIPTIC. 

TABLE  XLI3. 
COEFFICIENTS  FOR  £;  ARG.  N' . 


7     „    a    it    ^.t 

No. 

n' 

c 

D'c 

Be/Be 

dc/dr 

n'c 

i 

2 

O        I   —  I        O     O 
I        O  —  I        O     O 

0.0125 
1.00402 

—.0072 

+.0895 

-.0068 
+.0893 

—•131 
—  .0050 

—0.165 
+1.989 

—.0001 
+.0899 

3 

4 

2—1—1        00 
I        O        1—20 

1-995  56 
0.8464 

+.00247 
+.0033 

+.0025 

+.0098 

+.045 

.000 

+0.055 
+0.068 

+.0050 
+.0028 

It  should  be  added  that  the  coefficients  of  the  smaller  terms  show  only  the  order 
of  magnitude  in  each  case,  and  not  the  precise  numerical  value.  The  latter  will  be 
required  only  in  the  case  of  terms  found  to  be  sensible. 

The  theorem  that  all  the  inequalities  have,  as  a  coefficient,  the  motion  of  the 
ecliptic  during  nearly  one-sixth  the  period  of  the  argument  will  enable  us  to  limit 
the  combinations  of  the  arguments  to  be  considered. 

In  the  case  of  any  argument  N  ±  N'  containing  the  Moon's  mean  longitude, 
one-sixth  the  period  will  never  materially  exceed  5  days,  for  which  we  have 

-  =  o".oo64 

N 

In  none  of  the  terms  of  this  class  is  there  a  factor  C  so  large  as  to  bring  the  coefficient 
up  to  o."o5.  It  follows  that  all  the  combinations  N  =t  N'  which  contain  the  Moon's 
mean  longitude  may  be  omitted. 

Of  the  terms  which  remain  none  can  have  a  period  several  times  greater  than 
that  of  the  node,  for  which  the  ratio  K  :  N  =  i".5.  It  follows  that  no  combinations 
of  arguments  giving  products  of  coefficients  less  than  o.oi  need  to  be  computed. 

Numbering  the  arguments  N  as  in  the  first  column  of  the  tables,  these  two  rules 
will  be  found  to  leave  the  following  combinations  as  the  only  ones  to  be  considered: 

A;  +  Ay  =  e        jvt  +  Ay  =  2-*  -  0  A;  -  Ay  =  a/'  -  e  A;  +  Ay  =  2/'  -  0 

Using  these  numbers  the  computation  of  the  formulae  (43)  gives  the  following 
values  of  the  coefficients  £„,  kM  etc.,  for  the  argument  9. 

Arg. 

A;  -  AV 
A;-A? 

A;  +  A;' 

Sum 


h 

*. 

fc 

*a 

*. 

*Y 

.000358 

O 

+  0.0891 

+  0.1779 

—  O.OIO2 

+  1.972 

I 

O 

I 

+  .0002 

+  .0050 

+  .003 

0 

O 

oooo 

+  .0002 

.000 

I 

O 

+   .0002 

+  .0004 

+  oooo 

+  .013 

0 

O 

—  .0002 

oooo 

oooo 

—  .001 

.000359 

O 

+  0.0889 

+  0.1785 

—  0.0050 

+  1.987 

132  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

It  will  be  remarked  that  the  only  accented  coefficients  are  those  of  the  last  two 
lines;  and  that,  as  the  combined  argument  6  —  II  is  the  only  one  included  in  sum- 
mation, the  accented  and  unaccented  £a,  etc.,  may  all  be  combined. 

For  these  numbers  we  derive  by  (45)  and  (46) 

C,=  +  0.3157  C,  =  +  0.0506  C,  =  -  11.067 

Ca  =  +  0.001393         Ct  =  +  0.000187         c-,=  -  °-49S2 
We  have  from  the  adopted  elements  of  motion  of  the  ecliptic: 
*  =  o".47i4  n  =  173°  30'  +  o'.59(/  -  1850) 

The  following  are  then  the  results  for  the  argument  0: 


8/0  --  o".44i  cos  (0  -  II)        $7r0  =  -  o".oi  cos  (6  -  II)        80  =  +  i5".45  cos  (6  -  II) 

(4Q/) 
8a  =  -  o".ooi945  sin  (6  —II)     Se  =  o".ooo  By  =  +  o".6^i  sin  (0  -  II) 


To  these  expressions  for  8/0,  Sir0,  and  800  are  to  be  added  the  respective  increments 
fSndt          f&v^t          and         JX«7 

arising  from  substituting  the  values  of  8»(=  —  f«8a),  8e(=  o),  and  8y  in  the  ana- 
lytic expressions  for  n,  TT,,  and  0,. 

The  value  of  8a  gives  the  inequality  of  n 

BH  =  —  |w8ot  =  o".oo292«  sin  (6  —  II) 
This  adds  to  the  mean  longitude  the  inequality 

£/=  —  o".  002921;  cos  (6  —  II) 

where  v  is  the  ratio  n:(61  —  Z>(H)  =  —  248.7. 

The  complete  inequality  of  the  mean  longitude  thus  takes  the  coefficient  o".285. 
We  have  from  §  27,  (74) 

BTTI  =  (.02283  —  .  0043387)  n 

The  substitution  of  the  preceding  values  of  Sot  and  Sy  gives  the  increments 

&irl  =  —  .00304  sin  (6  —  H)         and         8?r  =  —  o".'j6  cos  (0  —  II) 
We  find,  in  the  same  way,  the  increment 

80=  +  o".n  cos(0-II) 

The  inequalities  of  /,  TT,  and  6  now  become 
87=  +o".  285  cos  (6-  H)     87r  =  -o".77cos(0-n)     80  =  +  is".s6  cos  (6  -  II)     (50) 

The  coefficients  of  the  arguments  2-rr  —  6  and  2/'  —  6  seem  so  small  that  we  leave 
them  out  of  consideration. 


NODAL  TERMS.  133 

§  65.  Inequalities  arising  from  the  coefficients  E  and  F. 

These  inequalities  have  been  considered  separately  on  account  of  their  minute- 
ness, and  on  their  depending  on  arguments  different  from  those  of  the  other  in- 
equalities. Some  special  values  of  the  coefficients  E  and  /''for  Venus  are  given  in 
tabular  form  in  Table  XII.  In  these  expressions  the  axis  of  X  passes  through  the 
mean  Sun,  as  in  the  case  of  the  inequalities  depending  on  the  mean  longitudes. 
But,  on  essaying  the  computation  of  the  principal  inequalities  arising  from  E  and 
F,  it  was  found  that  a  fixed  axis  of  X  would  be  more  convenient  to  use.  The  ex- 
pressions were  therefore  transformed  so  as  to  refer  them  to  the  Sun's  perigee  as  the 
initial  axis.  From  the  form  of  the  expressions  the  equations  of  the  transformation 
for  x  and  y  are  readily  found  to  be 

x'  =  x  cos  g'  —  y  sin^'  y'  =  x  sing-'  +  y  cos  g' 

where  the  accents  refer  to  the  fixed  solar  perigee.     It  follows  that  if 
E  =  a  cos  N  +  b  sin  N  F=  a'  cos  N+  b'  sin  N 

be  any  pair  of  the  terms  E  and  F  depending  on  the  argument  A,  the  correspond- 
ing transformed  terms,  which  we  represent  by  E'  and  F',  will  be 

E'  =  l(a+  b')  cos  (N+gr)  4-  1(«  -  b')  cos  (JV-gf) 

+  l(b-  a')  sin  (N+  g'}  +  %(b  +  a')  sin  (JV-  g') 

F'  =  \(a'  -  b)  cos  (N  +  g1)  +  \(a'  +  b}  cos  (IV  -g') 


The  transformed  expressions  thus  arising  are  shown  subsequently  in  Table  XLII. 
As  a  check  against  any  large  accidental  error  in  the  development  of  the  coeffi- 
cients, their  approximate  values,  neglecting  the  small  eccentricities  of  Venus  and 
the  Earth,  were  also  computed  by  analytic  development  as  follows:  Taking  the 
mean  radius  vector  of  the  Earth  as  the  unit  of  distance,  and  putting  a  for  the  corre- 
sponding numerical  expression  for  the  radius  vector  of  Venus,  the  Laplace-Gauss 
form  of  development  will  give 

A-5  =  J23w  cos  t'L 


L  being  the  difference  of  the  heliocentric  longitudes  of  Venus  and  of  the  Earth 
which  we  represent  for  the  present  by  /  and  /'  respectively. 

The  expressions  for  the  rectangular  geocentric  coordinates  of  Venus  will  then 
be,  when  powers  of  the  eccentricities  and  inclination  are  dropped  in  the  development 

X  =  —  cos  I'  +  a  cos  /         Y  =  —  sin  /'  4-  a.  sin  /         Z  =  a  sin  /sin  (/  —  0V) 

where  /  is  the  inclination  of  the  orbit  of  Venus,  and  6V  the  longitude  of  its  node, 
reckoned  from  an  arbitrary  fixed  origin.  Forming  the  product  of  the  several  fac- 
tors which  form  E  and  F,  noting  that  the  summation  changes  from  positive  to 


134  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

negative,  changing  and  transforming  the  indices  so  as  to  reduce  the  summation  to 
its  simplest  form,  the  values  of  E  and  F  take  the  following  general  form: 

-  a2£6fi+2))  sin  (iL  —  zl'  4-  0V)} 


sin  I 
F= 2{  -(a6f<*+l>  -  a2£6«>)  cos  (iL  +  <?v)  4-  W+1)  -  «2£6('+2))  cos  (iL  -  zl'  +  0V)} 

If  we  put,  for  brevity 

fi.  =  J(ot3e(l'+1)  —  «2£5(<))  sin  /         /8/  =  i(a^5('+l)  —  o23s(<+2))  sin  / 

we  shall  have 

E  =  2/3.  sin  (iL  4-0)4-  2/3.'  sin  (iL  —  zl'  +  0V) 

F=  -  2/34  cos  (iL  4-  0V)  4-  2/3.'  cos  (iL  -  zl'  4-  6>v) 

The  numerical  values  of  the  coefficients  b(i}  may  be  taken  from  any  one  of  various 
publications.  In  Astronomical  Papers  of  the  American  Epkemeris,  Vol.  V,  Pt. 
IV,  p.  343,  are  found  values  of  c5(i)=«2£(i)  for  Venus  and  the  Earth,  as  follows: 

*=     o  i  2  3  4 

c5("=  44.88  43.64  40.61  36.52  31.99 

From  these  we  find: 

i  =  —  2 
fi<56('-*-1)=  15.10 

ft -f- sin  7=    4.95 
ft'-*-sin/=    3.88 

We  thus  have  the  following  general  expressions  for  E  and  /%  the  axis  of  X,  in  the 
ecliptic,  being  arbitrary. 
We  use 

sin  /=  .0592 
Then 

E=  +  .293  sin  (—  zL  4-  0V)  4-  .229  sin  (—  zL  —  zl1  4-  0V) 
+  .273  sin    (-  L  4-  0V)  4-  .273  sin  (-  L  —  zl'  4-  <?v) 
4-  .229  sin  0V  +  .293  sin  (—  2/'4-  0V) 

4-  .186  sin  (L  4-  #v)        4-  -291  sin  (—  L  —  zl'  4-  0V) 
4-  .148  sin  (zL  4-  0V)      4-  -274  sin  (zL  —  zl'  4-  0V) 
F=  —  .293  cos  (— zL  4-  0V)  4-  -229  cos  (—  2Z  —  zl'  4-  0V) 

—  .273  cos  (—  L  4-  0V)   4-  .273  cos  (—  L  —  zl'  4-  0V) 

—  .229  cos  0V  4-  .293  cos  (—  zl'  4-  0V) 

—  .186  cos  (L  +  0V)       +  .291  cos  (L  —  zl'  +  0V) 

—  .148  cos  (zL  +  0V)     4-  .274  cos  (zL  —  zl'  4-  0V) 

Measuring  #v  from  the  solar  perigee,  in  longitude  279°. 5,  we  have 

0V  =  1550.4  P=g'  4-180° 


—  I 

o 

4-  i 

4-2 

15-53 

15.10 

14.05 

12.64 

10.91 

11.22 

10.91 

10.15 

4.62 

3.88 

3-H 

2-51 

4.62 

4-95 

4.92 

4.64 

NODAL  TERMS. 


135 


The  results  both  of  this  computation  and  of  the  analytic  development  are  shown 
in  tabular  form  as  follows: 

TABLE  XLII. 

E  AND   F  FOR    THE    ACTION    OF   VENUS. 


Arg. 

1 

? 

Arg. 

j 

F 

T          rfl 

CC 

s 

si 

! 

T            rrl 

C< 

>s 

s 

n 

L»    g 

Num. 

Anal. 

Num. 

Anal. 

L»  S 

Num. 

Anal. 

Num. 

Anal. 

o     o 

+.088 

-4-  OO7 

+•095 

.OOO 
+  008 

.OOO 

0       0 
O        I 

+.197 

-4-  oiJ. 

+.208 

.OOO 

—  .005 

.OOO 

O        2 
I   —2 

+.124 

+.I2O 

+.122 
+.121 

+.259 

—.252 
006 

+.266 
-.264 

O       2 
I    —2 
j         i 

—.255 
—.252 

+.Q2O 

-.266 
-.264 

+.116 
—.113 
+  on 

+.122 
—.121 

I         0 

+.183 

+.IQI 

+.078 

+.078 

I        O 

I        I 

+.398 

-4-  oio 

+417 

—.034 
008 

—.036 

I        2 
2  —2 
2  —I 
2        O 
2        I 
2        2 

+.U3 

+.112 
+.OIO 
+.174 
+.OIO 

+.096 

+.114 

+.114 

+"183 

+.095 

+.237 
-.236 
—.002 
+.129 
+.014 
+.198 

+.248 
—•249 

+Ti32 
+^208 

I        2 
2  —2 
2   —I 
2        O 
2        I 
2       2 

—28 

—.236 

+.022 
+.382 
+.006 
—  .200 

—.248 
—•249 

+401 
-^08 

+.104 
—.104 
-.008 
-.058 
-.008 
+.087 

+.114 
—.114 

-!o6o 
+.095 

The  largest  terms  arising  from  E  and  F  are  those  whose  arguments  are  inde- 
pendent of  the  mean  longitude  of  the  Moon,  Sun,  and  Earth.  These  arise  from  the 
constant  terms  of  E  and  F,  which  are,  when  referred  to  the  solar  perigee 


E=  +  .088 


=  +  .197 


The  computation  of  the  inequalities  arising  from  this  pair  of  terms  will  be  yet 
further  simplified  by  taking  the  node  of  Venus  as  the  axis  of  A.     By  transforming 

to  this  axis  we  shall  have 

.ZT=  .002  F— 


We  may  regard  this  value  of  E  as  evanescent,  thus  confining  the  terms  we  have 
to  determine  to  the  expression 


From  the  expressions  for  £,  77,  and  £  we  find  the  largest  terms  of  the  products 
and  their  derivatives  to  be : 

2££  =  —  .0895  sin  0  —  .0039  sin  (2/'  —  6)  +  .0006  sin  (ZTT  —  6) 
2Z>'^=  — .1786     "    —.0135  "  +.0012         " 

—^  =  —  .0137     "    +.0006          "  +.0218         " 

Be 


-  .  089 


« 


+  -014 


136  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

27;?=  +  .0895  COS  0  +  .0039  COS  (2/'—  0)  —  .OO06  COS  (27T  —  0) 
yrj%  =  +  .1786       "       +  .0135  "  —  .0012  " 


=  +  .0137     "     —.0006  "  —.0218  " 

Brit 


"     +.089  —.014 

The  resulting  terms  of  H,  heretofore  omitted,  are 


Taking  the  node  of  Venus  as  origin,  we  have,  as  shown  on  p.    135,  the  following 
terms  of  E  and  F 

E  =  .285  sin  2/'  F=  -.216+  -285  cos  2/' 

With  these  numbers  we  find  for  argument  6 

H  =  —  .0182  cos  0  D'H=  —  .0348  cos  0  -=r-  =  —  .0032  cos  0 

—  -  0  dJi-dJi-  5H  _ 

These  derivatives  are  to  be  substituted  in  the  fundamental  equations  (41)  and  (42), 
§  21,  and  each  equation  integrated.     For  the  latter  process  the  factor  of  integration  is 

j=-  248.8 
i 

The  product  of  this  into  M  for  Venus  (§  17)  is  —  i".o55 
We  thus  have  the  following  results: 

=  +  .o6iMcos  0  D^FQ  =  —  .o'joMcos  0  -A.A  =  —  2. 24^" cos  0 


S/0  =  —  o".o64  sin  0  8ir0  =  +  ".074  sin  0  80Q  =  -f  2". 36  sin  0 

~)  ,«  =  +  .oooi36.fl/"sin  0         D.e=  +  .ooood.'jMs'm0  D  .7  =  —  .loi^Ms'm  0       ' 

*t(  v  nt  it  nt  *  i 


Sa=  +  o".  000144  cos  0  Be  =  -f  ".000050  cos  0  By  =  —  o".io7  cos  0 

To  find  the  complete  inequalities  in  /,  TT,  and  6  we  must  add  the  respective  quantities 

f&ndt 


of  which  the  expressions  in  terms   of  8«,  8e,  and  8y  are  formed  by  §  27,  Eq.  74. 
We  thus  have,  dropping  unimportant  terms, 


=  —  !«8ot  =  —  o".ooo2i6w  cos  0          STT,  =  —  o.oi488w  —  o.oo43«87  =  +  o".ooo46w  cos  0 
&0l  =  +  .00388^  -f  .ooo66«&y  =  —  .00007  1  M  cos  ^ 


NODAL  TERMS.  137 

The  completed  values  of  8/,  8?r,  and  80  thus  become 

S/  =  -f  o".054  sin  0  +  8/0  =  —  o".oio  sin  6 

STT  =  —  0.115  sin  &  +  ^""o  =  ~  o".O4i  sin  6  (52) 

80  =  +  0.018  sin  6  +  800    =  +  2".38  sin  0 

in  all  which  expressions  6  is  reckoned  from  the  ascending  node  of  Venus. 

The  coefficients  of  the  term  in  2ir  —  6  are,  for  8y  and  80,  less  than  one  hundredth 
those  for  0,  and  the  integrating  factor  v  is  less  than  0.3  as  great.  The  coefficients 
in  2/'  —  6  are  but  a  fraction  of  those  in  0,  and  the  integrating  divisor  is  nearly  40 
times  as  great.  We  therefore  conclude  that  the  inequalities  depending  on  these 
arguments  are  inappreciable. 

§  66.  Action  of  Mars  and  Jupiter.  In  Mars  the  product  J/sin  /is  about  .08 
that  for  Venus.  I  have  therefore  not  computed  the  terms. 

In  the  case  of  Jupiter  the  largest  quantities  which  enter  into  the  constant  part 
of  F  are 


-^-=1.26  YZ  =  \a*ya\I  sin  7=0.0231 

Hence 

a'*jF=  1.26  x  .02310'=  +  .000103 

The  product  io3J//r  is,  approximately, 

For  Venus     —  o"-92  For  Jupiter     +  o".i7o 

The  inequalities  depending  on  6  are  proportional  to  this  product.  We  conclude 
that  the  inequalities  arising  from  the  action  of  Jupiter  may  be  derived  from  those 
of  Venus  by  multiplying  the  coefficients  by  —  0.185.  We  thus  have,  from  the 
action  of  Jupiter, 

8(9  =  -  o".43  sin  (0  -  0,)  87  =  +  0.020  cos  (0  -  6,)  (53) 

where  0j  is  the  longitude  of  the  ascending  node  of  Jupiter  on  the  ecliptic.  The 
inequalities  of  the  other  elements  are  unimportant. 

§  67.   Combination  of  terms  depending'  on  the  longitude  of  the  Moorfs  Node. 
The  inequalities  (49),  (50),  (51),  (52),  and  (53),  all  depending  on  the  same  argument 
0,  may  now  be  combined.     We  shall  do  this  for  the  two  epochs,  1800  and  1900. 
The  value  of  II  which  I  have  derived  in  Elements  and  Constants,  p.  186,  there 

called  Z',  is 

n  =  i73°29'.7  -f  S4'-4^  (from  1850) 


Taking  approximate  values  of  the  nodes  of  Jupiter  and  Saturn,  and  this  value  of  II, 

we  have 

1800  1900 

n       173°  2'  1  73°  57' 


138  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

We  take  the  nodes  of  Venus  and  Jupiter  as  constant,  using  the  values  for  1850 

0y  =  75°-3  9,  =  98°.9 

Carrying  these  values  into  the  inequalities  of  the  elements  in  question  and  combin- 
ing them,  we  find: 

SI  =  +  o".029  sin  0  —  o".27i  cos  0 
STT=  —  o.io  sin  0       +  0.80  cos  0 
50  =  +  2.55  sin  0       —  17.33  cos  0  •  •  •  (for  1800) 
80  =+  2.31  sin  0       —  17.34  cos  ^  ' ' '  (*or  1900) 
&y  =  —  o".  1 14  cos  0  —0.769  sin  0  •••  (for  1800) 
&y  =  —  0.103  cos  0    —  0.770  sin  0  •  •  •  (for  1900) 

§68.   Special   computation  of  the  Hansenian   Venus-term  of  long  period. 
The  following  are  the  planetary  and  lunar  arguments  whose  differences  make 

up  the  argument 

i8v  —  i6g-'  —  g 
of  the  term  in  question. 

Planetary  Lunar 

(1)  l8v    —  l&g-'  g —  2g' 

(2)  18       —  17  g  —    g' 

(3)  18     -16  g 

(4)  18-15  ff  +   g' 

The  coefficients  h^,  h^,  etc.,  are  computed  by  §§  22  and  23.  The  planetary 
coefficients  MK,  MC,  and  MD  are  found  in  Table  X.  The  lunar  coefficients  ap, 
etc.,  are  given  in  the  next  chapter,  Tables  XLVIII  and  XLIX.  For  the  argument 
g —  g'  we  change  the  signs  of  a,  e,  g  and  k,  as  given  for  the  argument  —  g-\-g'. 

I  have  not  computed  the  coefficients  for  the  argument^  —  2g'  believing  their 
effect  to  be  insensible.  Their  characteristic  is  ee'"1  =  .000050,  and,  in  the  principal 
term  of  17,  this  is  in  Brown's  theory  multiplied  by  a  factor  of  the  order  of  magnitude 
.04.  The  largest  planetary  coefficient  being  0.5  -=-  io3,  the  value  of  h^  will  be  of 
the  order  of  magnitude  i"  -=-  10°,  which  would  result  in  a  term  in  8/  of  the  order  of 
magnitude  o".o2.  Actually,  the  computation  shows  that  the  combinations  (2)  and 
(4)  are  also  much  smaller  than  (3). 

We  have  now  all  the  data  for  computing  the  coefficients  h^,  h^,',  etc.,  from 
the  formulae  of  §  22.  The  results  are: 

*„.'-- "-5597    -10"  A.,  .'  =  +  ".4880    -4-io6 

hec'  =  —    .0052    -4- io3  ke,'=+    .0047    -4- ios 

hliC'  =  —    .00081  -T-  io3  k,  /  =  —    .00094  "^  IC>3 

h,  c  =  +    .084      -T-  io3  k^>,'=  +    .095       -T-  io3 

The  coefficients  for  y  and  0  are  much  smaller,  and  are  omitted.  The  coefficients 
we  have  given  correspond  to  the  argument 

N—  Ar4  =  g  +  i6g'  —  i8v  =  A 


HANSENIAN  INEQUALITIES.  139 

of  which  the  annual  motion  is 

N  =  -  4747".8 
giving 

"  =  -  3649 
We  therefore  have  the  following  inequalities  in  IM  IT,  and  e 

S/0  =  —  o".oo3  sin  A  +  o".oo3  cos  A 
STT  —  +  o  .31  sin  A  —  o  .35  cos  A 
Be  =  —  o  .019  sin  A  —  o  .017  cos  A 

The  term  of  8/0  is  so  minute  as  to  be  unimportant.      For  the  term  in  the  mean 
longitude  arising  from  8»  we  have 


which  gives 

8/=  -  n".i8cos^  +  9".7Ssin^  =  i4".83  sin  (A  —  48°55'.2) 

It  will  be  convenient  to  use  the  negative  of  this  argument  in  order  that  its  motion 
may  be  positive.     We  shall  therefore  write 

8/=  i4".83  sin  (i8v  —  i6g'  —g+  228°5s'.2) 

where  v  is  the  mean  longitude  of  Venus  measured  from  the  earth's  perihelion. 

It  will  be  of  interest  to  compare  this  result  with  those  reached  by  other  investi- 
gators. The  following  are  arranged  in  the  order  of  time.  Putting 

Z,  the  mean  long,  of  Venus  —  that  of  Earth 

M=  i8Z  +  ig'  —  g 

and  reducing  all  results  to  the  mass  1-^408,000  of  Venus,  there  has  been  found,  for 
the  direct  action,  by 

Hansen*  8/  =  15".  34  sin  (M  +  229°.  2) 

Delaunayf  =  16  .34  sin  (M+  228  .5) 

NewcombJ  =  14  .80  sin  (M  -f  229  .5) 

Radau§  =  14  .14  sin  (M+  229  .o) 

Newcomb  (above)     =  14  .83  sin  (M  '+  228  .9) 

To  judge  the  precision  of  this  value  we  have  to  estimate  the  error  to  which  the 
development  by  mechanical  quadratures  is  liable.  The  circle  being  divided  into 
60  parts,  any  coefficient  which  we  have  taken  as  A18  is  really  the  sum  of  an  infinite 
series  of  which  the  first  two  terms  are  Aa  ±  A¥i.  We  have  dropped  all  the  terms 
after  the  first.  From  the  progression  of  the  coefficients  it  would  seem  that  the 

*  Tables  de  la  Lune,  p.  9.  t  Conn,  des  Temps,  1862,  App.,  p.  58. 

^Action  of  Planets,  p.  286.  \Inegalitts  Planitaires,  p.  113. 


140  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

ratio  A(  :  Ai+i  is  approximately  i :  1.26,  whence  the  ratio  Aw  :  A&  would  be  about 
250.  The  error  of  the  computed  term  may  therefore  well  be  ±  o".o6.  It  has 
been  only  as  this  work  is  in  press  that  the  author  has  looked  into  the  possible  effect 
of  the  slow  convergence  ;  and  while  it  seems  likely  that  the  error  entering  through 
the  coefficients  J*C  and  C  will  not  exceed  that  just  stated,  the  same  may  not  be  true 
of  the  coefficient  D. 

A  quantitative  estimate  of  the  correction  may  be  made  in  various  ways;  but 
the  author  is  unable  to  enter  upon  the  subject  in  the  present  work. 

It  is  also  to  be  noted  that  the  term  as  above  computed  contains  the  effect,  what- 
ever it  may  be,  of  the  mutual  perturbations  of  Venus  and  the  Earth.  A  separate 
computation  has  been  made  of  the  fundamental  numbers  due  to  these  perturbations, 
but  as  the  final  result  of  the  coefficients  amounts  to  only  a  fraction  of  a  second,  the 
computation  has  not  been  completed.  The  effect  being  included  in  the  computed 
term,  a  knowledge  of  its  amount  is  necessary  to  compare  the  result  with  that  reached 
by  the  ordinary  method  of  development. 

The  change  in  the  term  as  computed  is  too  minute  to  account  for  the  observed 
variation  of  long  period  in  the  Moon's  mean  motion.  As  the  period  of  this  varia- 
tion seems  to  be  nearly  the  same  as  that  of  the  inequality  under  consideration,  the 
question  naturally  arises  whether  the  effect  of  the  indirect  action  may  be  appreciable. 
This  being  the  most  important  question  in  the  lunar  theory,  a  computation  of  the 
principal  part  of  the  indirect  term  has  been  made.  The  result  being  altogether  un- 
important, it  seems  unnecessary  to  do  more  than  present  such  a  brief  statement  of 
the  method  as  will  enable  the  subject  to  be  taken  up  by  another  in  case  the  author's 
conclusion  is  not  well  founded.  The  required  perturbations  of  the  Earth  by  Venus 
are  most  easily  computed  for  the  case  in  question  by  using,  instead  of  the  Lagrangian 
brackets,  the  corresponding  functions  of  the  coordinates.  The  formulae  necessary 
for  the  purpose  are  found  in  Moulton's  Celestial  Mechanics,  p.  291.  The  eccen- 
tricities have  been  dropped  as  unnecessary,  and  attention  was  confined  to  the  longi- 
tude elements.  The  terms  dependent  upon  the  action  of  the  planet  on  the  Sun  are 
also  dropped,  being  appreciable  only  in  terms  depending  on  small  multiples  of  mean 

a'3 
longitude.     The  development  of  -g-  used  in  the  computation  is  that  in  Action, 

pp.  248-251.     The  result  for  the  indirect  action  is 

81=  +  o".O44  cos  A  —  o".036  sin  A. 
This,  being  added  to  the  terms  already  found,  gives  for  the  entire  term 

S/=  i4"-77  sin  (i8v  —  \6g'  —g  +  228°  54') 
which  is  the  definitive  result  of  the  present  investigation. 


RADAU'S  TERMS.  141 

§  69.  The  Radau  terms  oj  long  -period,  Radau  has  computed  certain  addi- 
tional terms  of  long  period  due  to  the  action  of  Venus,  with  the  following  results, 
the  arguments  being  reduced  to  those  adopted  in  the  present  work: 


Sv  =  +  o".i40  sin  (ITT  +g—  2ov  -f  19^-'  +  171°)  Per.  =  34^.8 

-f  o  .no  sin  (g  —  26v  +  29g-'  +  62°)  127.2 
+  o  .056  sin  (g—  2iv  +  2ig)  8.35 

+  o  .019  sin  (ir.+  g-  23V  -f  24^'  +  295°)  55. 

+  O   .016  sin  (TT  +  g—  I5v  +  ng'  +  219°)  71. 

—  o  .012  sin  (27r  —  g  +  24V  —  26g'  +  159°)  58. 
+  o  .012  sin  (g—  23v  +  24g-'  +  14°)  7.6 

+  o  .008  sin  (tr  —  Q+g—  23V  +  24^+  101°)  28.2 

+  o  .004  sin  (20-£--f  23V  -24^-+  183°)  42. 

+  O   .003  sin  (TT—  g+  2IV  —  2lg'  +  288°)  148. 

The  first  three  of  these  terms  are  the  only  ones  that  need  be  considered  for  the 
practical  applications  of  the  lunar  theory.  The  third  might  also  be  omitted,  but  is 
easily  computed  in  connection  with  the  first. 

For  all  the  terms  except  the  second  the  planetary  coefficients  A,  B,  C,  and  D 
may  be  derived  with  all  necessary  precision  from  the  special  values  of  these  coef- 
ficients given  in  Table  VII,  by  the  following  process.  Putting 


let  the  value  of  the  planetary  arguments  for  which  we  desire  the  coefficients  be 

N=  hL  +  kg' 

Recalling  that  the  720  special  values  of  each  coefficient,  say  A,  are  arranged  in 
12  systems  of  60  indices  each,  the  special  value  of  N~  corresponding  to  they'th  sys- 
tem and  the  index  i  will  be 

7Vy  =  6°  x  hi+  30°  x  kj 

We  may  mark  each  special  value  of  A  in  the  same  way.  The  values  of  the 
coefficients  Ac  and  A,  will  then  be  given  by  the  equations 


=  2  AU  cos  N^ 
360^4,  =  2  Afj  sin  JVitJ 

The  terms  of  A  for  the  special  argument  TV  will  then  be 

A  =  Ac  cos  N+  At  sin  TV 

In  most  cases  the  computation  may  be  simplified,  as  in  the  usual  method  of 
executing  periodic  developments,  by  adding  together  in  advance  the  special  values  of 
A  which  are  to  be  multiplied  by  the  same  sine  or  the  same  cosine.  Another  method 


142  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

may  be  used  in  computing  these  terms  by  the  developments  found  in  Action, 
Chapter  III,  §  18.  Some  modification  is,  however,  necessary  owing  to  the  circum- 
stance that  in  that  work  the  rectangular  coordinates  are  reckoned  from  a  fixed  axis 
passing  through  the  earth's  perihelion  or  the  solar  perigee,  while  in  the  present 
case  the  axes  pass  through  the  mean  sun.  It  is  therefore  necessary  to  use  the  ex- 
pressions for  the  geocentric  coordinates  of  Venus  referred  to  this  moving  axis,  a 
development  which  may  readily  be  made  from  the  special  values  already  given 
for  the  coordinates  of  Venus  and  the  sun.  It  is  necessary  to  transform  the  table 
so  that  the  arguments  shall  be  the  mean  anomaly  of  Venus  instead  of  its  mean 
longitude  because  the  development  for  A"5  which  are  tabulated  on  p.  25  of 
Action  have  the  mean  anomaly  of  Venus  as  an  argument. 

I  have  applied  the  method  of  development  from  special  values  to  the  first  term 
with  the  following  results: 

Planetary  Coefficients  for  Arg.  2OV  —  2ig'  '. 

AC  =  +  -03562  A.=  +  .00694 

Bc=-  .02999  Bt  =  -  .00583 

Cc  =  —  .00563  Ct  =  —  .001  1  2 

Dc  =  +  .00634  D.  =  —  -°32S4 

Ke  =  -f  .03280  Kt  =  +  .00638 

Ce  =  —  .00282  Ct  =  —  .00056 

The  lunar  portion  of  the  argument  is  equivalent  iD  —  g,  of  which  the  indices  in 
Table  XL  are  (—  i,  o,  2,  —  2).     From  the  numbers  in  this  table  we  find  for  the 

direct  action 

7r  —  2ov  +  19^  +  10°) 


IT  being  measured  from  the  earth's  perihelion. 

This  coefficient  is  less  than  that  found  by  Radau  ;  but  the  lunar  argument  is  one 
to  which  the  present  method  is  not  well  adapted  and  a  redetermination  is  desirable. 

None  of  the  other  Radau  terms  are  completely  computed  in  the  present  work. 
Such  computations  as  I  have  made  seem  to  indicate  even  smaller  coefficients  than 
those  found  by  Radau. 


CHAPTER  IX. 

PERIODIC  INEQUALITIES  IN  GENERAL. 

§  70.  For  convenience  we  mention  the  formulae  derived  in  Part  I,  giving 
them  the  special  form  adopted  in  the  actual  numerical  work.  We  recall  that  the 
combination  of  any  lunar  argument  N  with  a  planetary  argument  N±  gives  rise  to 
two  arguments  G,  N+N^  and  N— JV4.  For  each  argument  there  are  two  terms 
in  the  Dnt  of  each  of  the  elements,  one  a  cosine  term;  the  other  a  sine  term.  We 
represent  the  coefficients  of  these  terms  for  the  element  a  by 

&a,c,  ^«,.»  A*,c',  and  h^ 
with  a  similar  notation  for  the  remaining  elements, 

e,  7»  /»»  *•„>  and  eo 

except  that  the  coefficients  for  the  angular  elements  have  the  negative  sign. 

The  expressions  of  these  coefficients  for  the  direct  action  are  given  in  extenso  in 
Part  I  by  the  equations  (46),  (47),  (48),  (50),  and  (51).  For  the  indirect  action  the 
coefficients  are  given  in  (64)  and  (65),  but  we  may  use  the  equations  for  direct 
action  by  making  the  substitution  indicated  in  §  25  (66),  which  gives  the  expres- 
sions for  the  sum  of  the  two  actions. 

For  convenience  in  computation  the  coefficients  are  so  used  as  to  give  the  result 
in  terms  of  o".ooi  as  the  unit.  The  numerical  values  of  the  planetary  coefficients 
practically  used  for  the  purely  periodic  inequalities  are  these 

Kt'  =  ™\MKc  -  m*Gc)         CJ  =  io\(\MCc  +  </J        A'  =  ™3(MDe  +  «VJ 

with  corresponding  values  of  Kj,  Cs',  and  D,'. 

Since  each  combination  of  a  lunar  with  a  planetary  argument  gives  rise  to  two 
combined  arguments,  one  equal  to  their  sum  the  other  equal  to  their  difference, 
the  coefficients  relating  to  the  latter  are  distinguished  by  accents. 

The  numerical  values  of  the  planetary  coefficients,  as  derived  from  the  numbers 
of  Part  II,  and  just  defined,  are  shown  in  the  following  tables. 


i44 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


Values  of  the  Planetary  K' -Coefficients,  Combining  Direct  and  Indirect  Action. 

TABLE   XLIII. 
ACTION  OF  VENUS. 


Arg^ 

v,  g1 

K: 

c; 

D: 

*: 

c.f 

D: 

1—2 

—  0.61 

—  o'tf 

—  O.O2 

—  o!ii 

It 

.00 

—  0.46 

I    —    I 

+25.04 

—10.73 

+35-66 

+0.03 

—.01 

—  0.05 

I           O 

+  2.44 

—  0.52' 

+  2.31 

—  O.O2 

+.03 

—  0.04 

2  —  4 

-  0.68 

—  O.O2 

+  0.61 

—O.O6 

-.06 

—   0.12 

2  —  3 

+  6.53 

-  1-74 

+20.01 

+  I.2I 

—47 

—  4-33 

2—2 

—29-33 

+  8.77 

—52.76 

+0.25 

—.12 

—  0.27 

2—1 

—  349 

+  0.50 

—  3-29 

—  O.O2 

+.03 

—  0.08 

3-6 

—  0.31 

+  O.O2 

+  0.28 

—0.24 

+.02 

—   O.22 

3-5 

—  0.94 

+  0.15 

+  6.54 

—  0.61 

+.15 

—   5.29 

3—4 

+10.08 

—  3.25 

+  12.11 

+2.25 

-.69 

—   2.69 

3  —  3 

+  4-64 

—  1-99 

—  12.92 

+0.19 

—•03 

—  0.16 

4-6 

+  0.61 

—  O.2O 

+  0.85 

+0.53 

-.19 

—  0.76 

4  -  5 

+  0.68 

+  0.03 

—  2.19 

+0.05 

.00 

+  0.40 

4  —  4 

+  7-1  1 

—  2.38 

—  9-72 

—  O.O2 

+.01 

+  0.08 

5  -8 

+    O.I2 

—  0.04 

+  0.71 

+0.17 

-.05 

—  1.04 

5-7 

—   043 

+  0.16 

-  o.8S 

—0.36 

+.12 

+  0.78 

5  -6 

+   0.82 

—  0.23 

-  1.36 

+0.09 

-.06 

+  0.37 

5-5 

+   840 

—  2.43 

—  7-39 

6-9 

+   O.O2 

—  O.OI 

—  O.O2 

+0.03 

—.01 

+  0.03 

6-8 

+  0.18 

—  0.06 

—  0.16 

+0.14 

—.05 

+   O.I2 

6-7 

+  1.19 

—  0.28 

—  O.I  I 

+0.24 

-.06 

+   0.21 

6-6 

+  7.32 

—  1.86 

-  6.64 

o.oo 

.00 

0.00 

7-9 

+  0.18 

—  0.05 

—  0.16 

+O.I2 

—.04 

+  O.II 

7-8 

+  1.13 

—  0.24 

—   O.II 

-j-O.22 

-.05 

+   0.21 

7-7 

+  6.25 

—  1.42 

-  5-79 

0.00 

.00 

0.00 

8  -13 

—  0.0045 

+  0.003 

—  1.289 

+O.o66l 

—  .022 

—  15.92 

NOTE.  —  The  units  in  these  tables  are  o".ooi. 


PLANETARY  COEFFICIENTS. 


TABLE   XLIV. 
ACTION  OF  MARS. 


Arg. 
M.    g* 

*7 

c> 

D: 

K: 

CY 

D: 

I   —I 
I        0 

—  0.78 

+  0.18 

+0.13 
-0.31 

+  2.34 
+  0.32 

a 

—  O.OI 

+0.08 

+.01 
+.01 

II 

—  0.05 
—0.27 

2   —2 
2    —I 

—11.70 
-  0.45 

+3-97 
+0.38 

+17.04 

-11.08 

+0.34 
—0.63 

—.01 

+.24 

+0.26 

+8.80 

3  —3 
3  —2 

+  1.38 
+  0.8  1 

—0.43 

—0.25 

—  0.95 
—  2.44 

+0.16 
+0.89 

—  J02 

—•27 

+0.15 
+2.69 

4  —4 
4  —3 
4  —2 

+  0.66 
+  1.86 

—   0.12 

—0.17 
—0.60 

+O.OI 

—  O.2O 
—   2.Q4 
—  0.50 

+O.22 
+  1.84 

+0.59 

—.04 

-.64 

-•25 

+O.2I 

+3-05 
—4.90 

5  —4 
5  —3 

—  0.52 

+  0.04 

+0.18 

—0.02 

+  0.34 
—  0.19 

-0.68 
—0.47 

+.21 

+.15 

—0.48 
-1-73 

6  -5 
6  -4 
6  -3 

—  O.22 

+  0.04 
+  0.15 

+0.07 

—  O.O2 
—0.05 

—  0.03 

—   O.I  I 

+  0.69 

—0.33 
—0.74 
—0.08 

+.09 

+.25 

+.05 

—0.06 

—1-34 
+0.50 

IS  -9 
IS  -8 

+  O.I  I 

+  0.04 

—  O.O2 
—  O.O3 

—  0.13 

—  1-55 

—0.15 

O.OO 

+.04 

+.01 

—0.25 
—0.70 

TABLE   XLV. 
ACTION  OF  JUPITER. 


Arg. 

J,  S' 

K: 

c: 

*>: 

A'/ 

c-: 

D: 

+  1    -2 

—  4-25 

+  o'-72 

+   4-20 

—072 

+o."i9 

—  043 

+  1    -I 

-41.87 

+12.87 

+61.07 

—1.53 

+0.28 

—  1.67 

+  1        0 

+  1.45 

+   O.2I 

+  2.08 

+1.35 

—0.12 

—21.72 

+2  -3 

+   2.OI 

—   0.41 

-  1.98 

—  O.2O 

+O.O2 

+  O.IO 

+2   -2 

+30.81 

-  8.39 

-17.37 

+0.38 

—O.o6 

+  0.35 

+2    -I 

+  7-86 

-  2.88 

—  12.22 

+3.26 

—0.94 

+  5-29 

+2        0 

—  0.21 

—  0.04 

+  0.53 

—  O.I4 

—  O.OI 

+  0.17 

+3  —3 

+  4-53 

—  0.63 

—10.71 

—0^0 

+0.08 

—  0.39 

+3  -2 

—  0.15 

+  0.06 

+  0.18 

—5-93 

+1.67 

—  3-69 

+3  -i 

+  0.24 

—  0.08 

—  0.40 

—0.97 

+0.38 

-  1.67 

TABLE   XLVI. 
ACTION  OF  SATURN. 


Arg 

s,  g1 

K: 

c: 

D; 

K: 

c.' 

D: 

I   —I 
I        0 

—246 

+O.OI 

+o&3 
+ox>3 

+348 
—2.64 

—  0.03 

+O.OI 

+o'.oi 

—  O.OI 

—  0.03 
—0.50 

2   —2 

2    —I 

+1-33 
+0.66 

—0.33 
—0.23 

-I-S7 
—0.94 

O.OO 
+O.O2 

O.OO 
—0.01 

O.OO 

+0.05 

146 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


§  71.  The  lunar  coefficients  fall  into  two  classes,  one  determining  the  elements 
a,  e,  and  y  and  called,  for  brevity,  the  a-coefficients;  the  other  determining  /,  TT, 
and  0,  and  called  the  Z-coefficients.  Those  of  the  first  class  are  computed  by 
the  formulae  of  §  20  and  §  22  ;  those  of  the  second  class  by  the  formulae  of  §  23, 
Eq.  (50).  In  the  computation  we  write  k  for  £4. 

The  a-coefficients  are  the  nine  products  of  the  factors  a,  e,  and  g  defined  in  §  22, 
Eq.  43,  into  p,  y,  and  k. 

From  §  22,  (46)  to  (48),  it  will  be  seen  that  by  using  the  planetary  factors  in  the 
form  just  given  and  taking  the  a-coefficients 


a<7, 


etc. 


the  coefficients  of  the  terms  of  DRt(a.,  e,  and  y)  will  each  be  the  sum  of  three  prod- 
ucts of  two  factors  each.  But  the  quantities  we  actually  compute  are  the  values 
of  2§e  and  zSy.  We  therefore  double  the  coefficients  for  8e  and  Sy,  using 


2eg, 


and  gk 


We  have  also  multiplied  the  inequalities  of  TT  and  6  by  the  factors  2e  and  2y, 
required  to  reduce  them  to  inequalities  of  the  actual  longitude  and  latitude.  To  do 
this  we  take  for  the  nine  Z-coefficients 


L',  L", 


2cP",  eP 


t 


Each  of  the  coefficients  to  form  a  term  of  Dntlw  2eD,tlir0  or  2yDnl6t>  will  then  be 
the  sum  of  three  products  formed  by  taking  one  factor  from  one  of  the  Tables  XLIII 
to  XLVI,  and  the  other  from  Table  XLIX,  the  product  Z>'Z4  being  divided  by  2. 

TABLE   XLVII. 
DATA  FOR  a-coEFFiciENTs. 


Arg. 
***.* 

Arg. 

i       i'      i" 

a 

e 

g 

o      o    o      o 

0 

000 

0.0000 

oo.oooo 

00.0000 

O        I     O       O 

g' 

O       0       O 

o.oooo 

00.0000 

oo.oooo 

I   —  I     O       O 

g-ff 

I   —I        O 

+2.0529 

+19-137 

—00.0029 

I        O     O       O 

g 

I   —I        0 

+2.0529 

+19.137 

—  00.0029 

I        I     O       O 

g+e' 

I   —I        0 

+2.0529 

+19.137 

—  00.0029 

2       O     O       O 

2g 

2  —2       O 

+4.106 

+38.274 

—00.0058 

—  2  —  O     2  —  2 

2n—2g 

O       2        0 

—0.0602 

—38.307 

—00.0400 

—  I        O     2  —  2 

2D—g 

I        I        O 

+1.9927 

—19.170 

—  00.0429 

O        02—2 

2D 

2        O       O 

+4.0456 

—  00.0336 

—00.0458 

I        02—2 

2D+g 

3—1      0 

+6.098 

+19.103 

—00.0487 

O   —  I      2   —  2 

20—  g" 

200 

+4.046 

—00.0336 

—00.0458 

O         I      2   —  2 

20+g' 

2        O        O 

+4.046 

—00.0336 

—  00.0458 

—  I        O     2        0 

2\-g 

I         I    —2 

+1.978 

—19-175 

+11.0971 

O        0     2        O 

2\ 

2        O  —  2 

+4.0306 

—  00.0388 

+11.0942 

1020 

2\+g 

3  —i  —2 

+6.083 

+19.098 

+11.0913 

O        O     O        2 

2\' 

O        O  —  2 

+0.015 

—  0.005 

+11.140 

VALUES  OP  THE  LUNAR  COEFFICIENTS. 


TABLE  XLVIII. 
LUNAR  H-COEFFICIENTS  FOR  a,  e,  AND  7-. 


Arguments. 

a/ 

zq 

y^k 

e# 

e? 

y^k 

st 

g? 

y*& 

g,  g'^  * 

/,    1C,      9,    g' 

o     o     o     o 

0       O       O       O 

0 

o 

o 

o 

o 

0 

o 

o 

o 

I        O       0       O 

+  1    —  I        0        0 

—0.032  09 

—.11147 

—0.034  8  1 

—0^299  i 

—1.039  i 

—0.324  6 

+.00005 

+.00016 

+.00005 

2        0       O       O 

+2—2        0        0 

—0.005  13 

+.00197 

—0.004  99 

—0.047  84 

+0.018  37 

—0.046  50 

+.OOO  OI 

o 

+.OOOOI 

—  I         I        0       O 

-I    +1        0   +1 

+0.000  13 

—.00073 

+0.00008 

+0.001  24 

—0.006  79 

+0.00077 

o 

o 

o 

I        I        O       O 

+  1    -I        0   +1 

—0.001  39 

+.00056 

—  0.001  38 

—  O.OI2  92 

+0.005  26 

—0.01282 

o 

o 

o 

—  r     020 

+  1    +1    —2        0 

—o.ooo  04 

—.00032 

+O.OOO  IO 

+0.000  38 

+0.003  17 

—0.00096 

—.00022 

—.00178 

+.00056 

O        O        2        O 

+2        O   —  2        O 

—  o.ooo  77 

+.024  10 

—0.00077 

+O.OOO  OI 

—o.ooo  23 

+0.000  01 

—.002  II 

+.06634 

—.002  1  1 

I        O        2        O 

+3    —I    —2        0 

—0.00006 

+.00199 

—0.00006 

—o.ooo  19 

+0.00630 

—  0.000  19 

—.000  1  1 

+.00366 

—.000  1  1 

—2        O        2   —  2 

0+2        O  —2 

—o.ooo  235 

—.000  054 

—o.ooo  216 

—O.I49  20 

—0.034  28 

—0.13790 

—xioo  16 

—.00004 

—.000  14 

—  I        O        2   —  2 

+  1   +1        0  —2 

—0.16243 

—.018  53 

—0.162  95 

+1.562  6 

+0.1783 

+1.5675 

+.003  So 

+.00040 

+.003  51 

O        O        2   —  2 

+2       O       O  —  2 

+1.9949 

—.028  26 

+1.99470 

—  0.016  56 

+0.000  23 

—  0.016  56 

—.02258 

+.00032 

-.02258 

I        O        2   —  2 

+3  +1        0  -2 

+0.1660 

—.001  28 

+0.1660 

+0.519  98 

—  0.004  01 

+0.51998 

—  .00132 

+.OOOOI 

—.00132 

0        I        2—2 

+2        O        O  —  I 

—0.00643 

+.00006 

—0.006  41 

+0.000  05 

35  •*•  10" 

+0.00005 

+.0^007 

68y-!-io» 

+.00007 

O  —  I        2—2 

2        O        O   —  3 

+0.00651 

—.00012 

+0.006  35 

—0.00005 

708  -s-  io9 

—  0.000  05 

—  .oo<  >  07 

i37-;-io8 

—.00007 

O        O        0        2 

O       O  —  2  +2 

+297-t-io7 

—  63  -no1 

—  297-j-io7 

—  99-t-lo' 

+  2I-S-IO7 

+99-*-io7 

+.02206 

—.00468 

—  .02206 

TABLE  XLIX. 

LUNAR  Z.-COEFFICIENTS  FOR  /,  n,  AND  0. 


Arguments. 

L' 

L" 

L> 

*eP' 

2eP" 

eP, 

ifR 

2rJ?" 

r^ 

g,  s'  i,  i 

/,  ff,  e,ff' 

o     o     o     o 

I        O       O       O 
2       O       O        O 
—  I        I        O       O 
O         I        O        O 
I        I        O       O 
—  I        O        2        O 
O       O       2       0 
I        O       2        0 
—2        0        2—2 
—  I        O        2—2 
O        O        2   —  2 
I        O        2   —  2 
202—2 
—  I         I        2   —  2 
O                  2—2 
I                   2—2 
—  I    —           2   —  2 
0   —           2—2 
I    —           2—2 
—  I        O        O        2 
O        O        0        2 
I        O        O   —  2 
—  2   —  I        2   —  2 
—2         I        2   —  2 

o     o     o     o 

+  1    —  I        0        0 
+2    —  2        O        O 
-I    +1        0        4 
O        O        O   +1 
+  1    -I        0  +1 
+  1    +1    —2        0 
+2        O  —2        O 

+3  —I  —2      o 

0+2+0—2 
+  1    +1        O  —2 
+2         O        O   —  2 
+3    —I        0  —2 
+4    —I        0   —2 
+  1    +1        0   -I 
+2        O        O  —  I 

+3  -i      o  -i 
+  1   +1      0  —3 
2        O       O  —  3 

—  0.08204 

—0.123  05 
—  0.010  05 
—  0.000  32 
—0.005  454 
-0.005  59 
—0.00009 
—0.00099 
—o.ooo  16 
+0.013  67 
—0.302  23 
+1.09222 
+0.10262 

+0.005  12 

—0.003  614 
—0.00922 
—  0.001  58 
+0.001  810 
+0.00943 

+O.OO2  27 
—O.OOO  1  1 

+0x05  65 
—0.00037 
—o.ooo  019 

+O.OOO  IOI 

+2.004  31 
—0.199  14 

+O.OO2  48 

—0.003  16 
+0.001  241 

+O.OO2  42 
+0.000  1  1 

+0.01775 

+0.001  OS 

+0.00684 

—O.070  22 

—0.076  97 
—  0.002  34 

—  O.OOOII 

—o.ooo  172 
+0.000  94 

+0.000  OI 

—0.003  176 
—0.004  14 
—o.ooo  15 
—0.00007 
—o.ooi  69 
—0.000  95 
+0.000  130 
—  0.000006 

—0.11635 
+0.595  03 
+0.077  07 
o 
—0.004  743 
+0.02651 

—0.181  96 
+2.05609 
+0.057  30 
+0.006  73 
—  o.ooi  172 
—0.00039 
+0.024  25 
+0.003  15 
—0.008  24 
—0.06908 
+0.36088 

+O.O22  69 
+O.OO7  90 

—  GO"?  8^ 

+•135  12 
—.012  85 

—0.27507 

—0.019  74 
+0.002  44 
—0.014  128 
—0.01086 
+0.00038 
—0.00203 
—o.ooo  32 

+0.02351 

—  0.612  74 
+3-984  14 
+0.205  24 
+0.010  24 
—0.007  273 
—  0.018  40 
—0.003  16 
+0.003  66  1 
+0.01866 
+0.004  54 

—  O.OOO  22 

—  o.on  29 

+0.00090 

—o.ooo  207 
+0.000  224 

+0.6404 
+0.072  8 

—.00246 
—.000  oi 

—.00250 
—.00004 

—0.00653 
+0.027  o 

—.000073 

+.000041 

—.00012 

+.014  50 

—•13790 

—.00490 

+XXJOOI 

—.001  36 
—.010  28 

+O.OOO  02 

+O.O0002 

+.00433 

+.00378 

—0.29875 
+3-11720 
+0.298  oo 
—  1.027  38 
—0.11462 
+0.015  989 

+0.000  02 

+0.006  73 
—0.009992 
—  o.ooo  02 
—0.008  927 

-0.275  19 
+3-1  17  4 
+0.2981 
—  1x1273 
—0.1146 
+0.015  785 

+XWOO4 
—.00500 
+.045  17 
+.00180 

+.OOO  OI 

—.000003 

+.00004 
—.005  03 

+•045  14 
+.001  75 

+.00001 

—.000002 

+0.0063 

—  O.OIO  I 
—  O.OOOO2 
—0.0088 

+O.OI5  462 
+O.OOOOI 
—0.00464 

+.000  002 

—XXX)  OO3 

+.000  ooi 

3  —  i      o  —  3 

—I    +1    —2   +2 
0        0—2+2 
+  1    —I   —2  +2 

—  0.000  23 

+0.000  04 

+O.OOO  22 

-.04875 

+.00885 

+.04880 

+0.000  915 
—  0.002  797 

—0.002766 
—0.000042 

+0.002  976 
—0.002966 

o 
o 

o 
o 

o 
o 

148  ACTION   OF  THE   PLANETS  ON  THE   MOON. 

§  72.  From  these  two  tables  the  four  coefficients  for  each  element  are  formed  by 
the  following  computation,  an  adaptation  of  (46)  to  (51) 

The  inequalities  of  e  have  received  the  factor  2,  and  those  of  TT  the  factor  2e  in 
order  to  transform  them  into  the  principal  terms  of  the  true  longitude  without 
further  multiplication. 

Two  other  points  which  may  be  recalled  are  these:  (i)  We  use  k  instead  of  Kt 
in  the  formulae;  (2)  it  is  to  be  recalled  that  Cc'  and  C,'  contain  only  ^C,  as  that 
symbol  is  used  in  Part  I. 

Element  a. 
«,  =  *>.#  -  C,'ag  «2  =  -  JT'a/  +  C^q 


Element  e 
y  *,  =  -  2/T/e/ 


Element  y 


Element  /0 
\  =  Kc'L'-Cc'L"  \  =  Xi 

hlt  c  =  \-  \D:L,  hlt  /  =  x, 

+  \  *i.  /  =  i 

Element  TTO 


-  \ 


-  2  C'eP" 


=  TT,  -  D.'ePt 
=  Dc'ePt  +  7r2 


Element  60 


In  the  exceptional  cases  when  one  of  the  constituent  factors  of  either  class, 
planetary  or  lunar,  is  a  constant,  there  will  be  a  merging  of  the  accented  and  unac- 
cented arguments  and  terms. 


PERIODIC   INEQUALITIES   IN   GENERAL.  149 

For  the  case  JV  =  o,  a,  e  and  g  all  vanish,  and  we  have 

A(«  =  A/  =  A,7  =  o  £  =  o 

while  (49)  of  §  23  may  be  written 

-  DJ.  =  (tK.'LJ  -2Cc'La")  cos  7V4  +  (iK.'LJ  -  2C.'L9")  sin  Nt 
We  have,  therefore,  in  this  case,  only  to  double  the  values  of  the  Z-coefficients 
for  argument  o. 

In  the  combination  of  a  constant  planetary  factor  (7V4  =  o)Vith  a  periodic  lunar 
factor  we  may  use,  instead  of  (46) 


Then 

Sa  =  —  vh^  ,  cos  N 

with  similar  equations  for  e  and  y,  formed  by  writing  e  and  g  for  a.     We  also  have, 
instead  of  (51) 


Then 

8/0  =  -  vhlc>  sin  N 

with  similar  equations  for  IT  and  6. 

As  neither  D  nor  /has  a  constant  term,  there  are  only  cosine-terms  of  this  class 
in  a,  e,  and  y,  and  only  sine-terms  in  /,  TT,  and  0. 

From  these  coefficients  for  the  Dnt  of  the  elements  we  have  those  for  the  ele- 
ments themselves  by  multiplication  by  the  integrating  factor  v.  The  motion  of  the 

lunar  argument  is 

in  +  i'irl  +  t"0l  +  jn'  =  N 

and  that  of  the  planetary  argument 

k'n'  +  knt  =  N4 
We  compute 


«  I 


v  = 


Then  the  coefficients  which  we  compute  are 

«.  =  Vka,c  «„  =  -  "/'„,,  «.'  =  "'^«,  „'  «/   =  -  V'kJ 

2e,'  =  2v'hetC'  2ec'  =  -  2v'/iti,f 


27,  =  2v/lyiC  2JC  =  —  2V/ly>,  2y,'  =  2v'kyc' 


rs  =  —  i/  x 
Tr  =  -f-  1/  x 

with  similar  forms  for  0  when  required, 


150  ACTION  OF  THE  PLANETS  ON  THE  MOON. 

The  inequalities  of  the  elements  are  then 

SI  =  lc  cos  (N+  N,}  +  1.  sin  (N  +  N^  +  //  cos  (N-  JVt)  +  //  sin  (N  -  1VJ 


STT  =  7TC  "  +   TT.  "  +7T/  "  +  IT.'  " 

$0  =  ec  "  +  e.  "  +  <?„'  "  +  e.'  " 

A  similar  computation  was  made  for  y  and  0;  but  the  results  were  unimportant  in 
all  but  one  of  the  arguments. 

§  73.  The  motions  of  the  arguments  from  which  the  integrating  factors  v  or  v 
are  to  be  computed  are  the  following.  The  sidereal  motion  for  a  Julian  year  is 
given  in  revolutions  for  the  lunar,  and  in  seconds  for  the  planetary  arguments. 
Then  follows  the  ratio  of  each  to  the  mean  motion  of  the  Moon. 

Motions  of  Arguments. 

Mot.  in  365d.25  n 

£•;£".=  i3'-255  523  0.9915452 

/;    «=  13  .368  513  i. 

TT  ;?!•,=  0.112990  0.0084518 

0  ;   01  =  —   o  .053  765  —  0.004  0218 

Venus  2106  64i".38  0.121  5913 

Earth  1295  977  .43  0.074  8013 

Mars  689  050  .9  0.039  77°7 

Jupiter  109  256  .6  0.006  3061 

Saturn  43  996  .2  0.002  5394 

The  elemental  inequalities  computed  from  these  formulae  are  shown  in  tabular 
form  on  the  following  pages.  On  making  the  computation  it  was  found  that  the 
coefficients  for  a  were  so  minute  that  no  terms  in  the  parallax  would  need  to  be 
considered,  and  only  in  some  exceptional  cases,  generally  terms  of  long  period,  did 
the  inequality  of  y  affect  the  longitude.  The  coefficients  for  these  elements  are 
therefore  omitted  in  the  tables  of  longitude  elements.  The  given  coefficients  are 
those  for  the  mean  longitude,  8/,  280,  e8ir.  It  must  be  remembered  that  the  accented 
e'  and  ir'  do  not  refer  to  solar  elements,  but  designate  only  the  coefficients  depending 
upon  the  differences  between  the  lunar  and  the  planetary  arguments,  while  the 
unaccented  coefficients  depend  upon  their  sum. 

It  was  also  found  that  the  inequalities  of  y  and  6  were  insensible  in  nearly  all 
cases.  The  few  terms  of  these  elements  found  to  be  sensible  are  therefore  given 
separately. 

§74.  Terms  -with  purely  Lunar  Arguments.  We  here  make  a  single  com- 
putation for  the  combined  action  of  all  the  planets.  To  include  the  effect  of  the 
indirect  action,  we  have  only  to  modify  the  values  of  MK,  etc.,  as  indicated  in  (66). 


INEQUALITIES  OF  ELEMENTS.  151 

Then,  from  the  values  of  the  constant  term  already  given  for  the  four  principal  dis- 
turbing planets  in  §  54  we  find 

io'2y]/^T0  =  +  6".o7o  iosI.MC0  --  5".76 

-  io3w2£0  =  -  0.459  loW/,  =  +  0.153 

lo3/^'  =  +  5.611  io3C0'  =  —  2.727 

For  the  terms  in  question  we  now  have,  for  each  lunar  argument 


and 


the  terms  in  y  and  6  being  omitted  as  unimportant. 

The  inequalities  of  /0,  e,  and  IT  may  now  be  computed  as  in  §§  26  and  27.     The 
most  condensed  formulae  of  computation  are 

ioV  -- 


The  elemental  inequalities  then  are 

SI  =  /t  sin  TV  e&ir  =  eiri  sin  TV  Se  =  ee  cos  TV 

The  results  of  this  computation  for  the  only  terms  which  I  have  found  to  give 
any  appreciable  result  are,  in  units  of  o".ooi; 

Arg. 

g 
?,D—2g 

2Z>* 

The  only  corrections  of  the  true  longitude  to  be  considered  are  the  following 
to  the  evection  and  variation. 


I03/. 

I0»ec 

icfeir. 

3-8 

-    9.0 

-    9.0 

+    1.5 

+  14.0 

-I3.6 

+      8.2 

+  21.6 

-21.5 

—  22.  0 

—     O.I 

-    0.9 

+      0.4 

+      2.O 

+      2.0 

sn    2      — 


o".O2i  sin  2D. 


§  75.  Elemental  Inequalities.  The  miscellaneous  inequalities  of  the  mean 
longitude,  the  eccentricity  and  the  perigee,  as  given  by  the  preceding  formulae  and 
data,  are  tabulated  in  the  following  pages. 

It  may  be  repeated  that  the  mean  longitudes,  v,  M,  j,  and  s,  are  measured  from 
the  solar  perigee. 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 

Periodic  Elemental  Inequalities  in  Units  of  o,"ooi. 
TERMS  INDEPENDENT  OF  THE  LUNAR  ARGUMENTS.     (1V=  o.) 


Action  of  Venus. 

Action  of  Mars. 

Arg. 

V 

I. 

I. 

2<?7Tt 

1<?7T. 

Arg. 

V 

I. 

'. 

2«r. 

2«r. 

V 

+      8.22 

-  14 

—    i 

+    6 

O 

M 

+  25.1 

—  30 

—     i 

+  4 

—  o 

v-/ 

+  21-37 

-831 

+    i 

+208 

0 

M-Z- 

—  28.6 

—  II 

+     I 

+  7 

0 

V—  2/ 

—  35-70 

+  56 

—    i 

0 

—   I 

2*1—  g" 

+2II.O 

+308 

-181 

-48 

+49 

2V—  2^ 

+  10.69 

+324 

+    5 

—107 

—   I 

2VL—2& 

-    14.27 

—200 

0 

+60 

0 

2V—  3^ 

+  53-25 

-314 

+  90 

+114 

-24 

3M—  2g 

—  33-o 

+  29 

-  3i 

—  9 

+  10 

2V—  4^ 

-  17-85 

+    3 

—    4 

+    3 

O 

3M-3^ 

—    9-5 

+  14 

0 

—  4 

o 

3V—  3^ 

+      7-12 

-  5i 

+     i 

+  13 

o 

4M—  2g' 

+  105-5 

+      2 

+  95 

3 

-24 

3V-4S' 

+  15.25 

—173 

+  36 

+  54 

—  12 

4M—  3^ 

-  15-3 

+   32 

—  35 

—  10 

+  10 

SV-Sg" 

-108.3 

-48 

+  54 

+  30 

—21 

SM—  3g' 

—  39-1 

+  23 

—  20 

—  i 

-  6 

SV-Sg" 

+105 

—  15 

+  18 

+    4 

-  6 

ISM-Sg' 

—540.5 

+  39 

+   22 

—  9 

—   2 

8v-i3/ 

+310-4 

+  16 

+246 

-    5 

-74 

Action  of  Saturn. 

Action  of  Jupiter. 

Arg. 

V 

I. 

'. 

a«r. 

2£/r. 

Arg. 

V 

t. 

1. 

2eng 

2£7rc 

S 

+  39-4 

+  48 

+  16 

—    4 

o 

J 

+1586 

+  171 

+  41 

+  4i 

—57 

s-s- 

-  13-8 

—  40 

0 

+  ii 

o 

1-* 

-  14.6 

—652 

+  13 

+2IO 

—  7 

25—  g' 

—  14-3 

+  " 

—    i 

-    3 

0 

1-of 

—    7-0 

—  15 

+    4 

+    9 

—   2 

25—2? 

-    6.9 

+    8 

o 

-    3 

o 

2}~g' 

—  16.1 

+165 

-  52 

-46 

-18 

2}—2g' 

—    7.30 

+208 

—     i 

-  74 

—  i 

2J—3g' 

—    4-7 

+    6 

o 

—    3 

o 

3J-S1 

—  17-9 

+    5 

—  24 

—      2 

-  6 

31—  2g" 

-    7-6 

—      2 

+  42 

+      I 

-15 

3J—3S" 

—    4.9 

+    9 

+  I 

-    6 

+  i 

LUNAR  ARGUMENT  N=. 


Planetary 
Argument. 

V 

I/ 

/. 

I, 

/; 

c 

»«. 

a*. 

*;. 

2Cc> 

2  ex, 

2<?'Tc 

2C7T,' 

2exc' 

V-/ 

+0.9631 

+  1.0585 

+     I 

—  O 

+16 

o 

o 

-13 

o 

-63 

—13 

0 

-63 

o 

2V-3^ 

+0.9898 

+  I.028I 

—    4 

+1 

+  6 

+  I 

+  i 

+  5 

+  5 

—  22 

+  5 

—  I 

—  22 

-  5 

2V—2gr 

+O.92I6 

+I-II37 

—     I 

o 

—  22 

o 

o 

+  i 

o 

+77 

+  i 

o 

+77 

o 

3V—  Sg" 

+  1.0180 

+0.0993 

+     i 

+1 

+    I 

+  1 

+  3 

+  5 

+  3 

-  3 

+  5 

-  3 

3 

^_    ij 

3V-4S" 

+0.9460 

+1.0800 

o 

o 

+  4 

o 

o 

-  5 

+  4 

—23 

—  5 

0 

-23 

—  4 

3V—  3^ 

+0.8814 

+I.I750 

+    3 

o 

I 

o 

o 

-13 

o 

+  i 

—  13 

o 

+  1 

O 

2M—  2g" 

+1.0853 

+0.9420 

-    6 

o 

+  i 

o 

o 

+29 

o 

+  5 

+29 

0 

+  5 

o 

2M—  g' 

+  1.0038 

+  1.0134 

+      2 

o 

—   2 

tj 

-  6 

+  9 

—  I 

o 

+  7 

+  5 

+  9 

+  5 

J-f 

+  1.0834 

+0.9434 

—   23 

o 

O 

o 

+    2 

+98 

—  I 

+  10 

+97 

—   2 

+  11 

o 

J 

+  I.OO22 

+1.0150 

O 

+5 

O 

+13 

+  5 

o 

+15 

—  2 

0 

—13 

—   2 

-IS 

2J—2g' 

+  I.I703 

+0.8861 

+  150 

o 

+40 

o 

o 

-54 

o 

—21 

-54 

0 

—21 

o 

2J—g' 

+  I.076I 

+0.9490 

+  40 

—  I 

oo 

-  8 

-  8 

—21 

+  i 

—  3 

—21 

+  8 

—  3 

—   I 

INEQUALITIES  OF   ELEMENTS. 
LUNAR  ARGUMENT  N=  iD  —  2g  =  in  —  ig'  (EVECTION-TERMS). 


'53 


Planetary 
Argument. 

V 

i/ 

I, 

1. 

// 

V 

2e, 

2C, 

**: 

«*; 

2  en. 

2ene 

2671,' 

•*/ 

V-g' 

—  11.64 

—    5-572 

—  i 

0 

+    6 

o 

o 

-  19 

o 

+   101 

+  18 

O 

—  101 

o 

2V—  3^ 

-    8.779 

-    6.602 

—   2 

o 

+     3 

+  I 

^_     ty 

—  30 

—  II 

+    50 

+  30 

-  8 

-     51 

—  ii 

2V—  2^ 

-  2S-S7 

—    4419 

+  9 

o 

-    6 

o 

O 

+  134 

__        J 

—  106 

-130 

o 

+  107 

—    i 

3V—  Sg' 

—    7-046 

—    8.100 

—  i 

+  I 

+    i 

+  I 

—  12 

-  IS 

—   IO 

+      12 

+  IS 

—  12 

—    13 

—  ii 

3V—  4^ 

—  14-90 

—    5-045 

0 

0 

+     2 

o 

O 

—     i 

—    7 

+    33 

o 

—   I 

—    33 

—    7 

3V—  3g" 

+  129.9 

-    3-66 

+72 

o 

0 

0 

—  2 

-661 

0 

—      7 

+659 

—   2 

+      7 

o 

6v-8/ 

—625.0 

-     3-79 

+50 

—37 

0 

0 

+49 

+  64 

0 

0 

-64 

+49 

o 

o 

SV-8^ 

+737-7 

—    3-43 

o 

o 

o 

o 

-  5 

-  24 

o 

o 

+  24 

-  5 

o 

0 

H-* 

-    5-062 

—  10.24 

o 

o 

0 

o 

o 

-    5 

o 

+      4 

+    5 

0 

—       4 

o 

2M—  2g" 

—    4-932 

-  15-96 

—  2 

o 

+    i 

o 

+  i 

-  42 

o 

+     IS 

+    42 

o 

—     15 

0 

2M—  g" 

-    7-815 

-    7-276 

+    I 

—   2 

—    5 

—     2 

+  17 

+  25 

+  19 

—    25 

-  25 

+  17 

+     25 

+  19 

3M—  3g' 

-    4-205 

—  36-23 

O 

0 

o 

0 

o 

+    3 

o 

+     6 

_         .j 

o 

—      6 

0 

3M—  2g' 

-    6.135 

-    9-766 

o 

o 

0 

O 

+  6 

+    6 

+    4 

—      4 

-    6 

+  6 

+      4 

+    5 

4M—  3^ 

—    5.050 

-  14.84 

o 

0 

0 

O 

+  7 

+    7 

+    4 

—      3 

—    7 

+  7 

+      4 

+    5 

4M—  2^ 

—    8.116 

•    7-034 

0 

0 

0 

o 

+  9 

+     I 

—  ii 

—      i 

—    i 

—  IO 

+      i 

—  ii 

iStf-Sg- 

—    7429 

-    7.64S 

o 

o 

o 

o 

—  I 

+    3 

—     i 

—      3 

—    i 

—29 

o 

o 

6M-5S7 

—    3-73 

+3704 

o 

0 

—    i 

-    8 

—  i 

0 

—  22 

+    29 

+    o 

—  i 

—    29 

—  23 

J—  2^ 

-    3-6232 

+  94-3396 

o 

0 

o 

—    i 

—  i 

-    9 

—   IO 

+     15 

+    9 

—  i 

-     16 

—    9 

J-S' 

—    4.9702 

-  15.5763 

-  8 

o 

+     3 

0 

—  5 

-151 

O 

+    54 

+I5i 

—  5 

-    53 

o 

J 

—    7-9II4 

-    7-1942 

o 

+   2 

O 

+    3 

—44 

—     I 

-46 

+      7 

+    i 

—44 

-      7 

-  46 

2J—  3^ 

—    2.9028 

+'  12.6422 

o 

o 

o 

O 

o 

+    3 

—      I 

—      i 

-    3 

O 

o 

—    I 

2J—  2^ 

-    3-7078 

+232.720 

+  3 

o 

+256 

+  I 

+  i 

+  54 

+    5 

-1158 

-  54 

+  I 

+1164 

+    3 

2J—g" 

-    5.1308 

—  14.1844 

+   2 

—  I 

—     I 

o 

+13 

+  30 

+    6 

—      12 

—  31 

+13 

+     IS 

+    6 

3J-3/ 

—    2.9568 

+  13.7363 

+  I 

o 

+  I 

o 

+  11 

+  13 

o 

+      22 

—  13 

—  I 

—      21 

0 

3J-2^ 

—    3.7965 

—497.760 

o 

+  I 

-    8 

-258 

—  ii 

o   +429 

—      14 

o 

—  II 

+     15 

+43i 

SJ-g' 

—    5-3022 

—  13.0208 

o 

o 

o 

o 

-  4 

+     i 

—      2 

0 

—    i 

—  4 

+      i 

—      2 

LUNAR  ARGUMENT  N=  2D  —  g=g'+  2n  —  2g-'. 


Planetary 
Argument. 

V 

!/ 

I, 

/ 

// 

V 

2*. 

2ge 

M.> 

«•/ 

2  en, 

2enc 

2  en,' 

2enc' 

V-g' 

1.1042 

I-23I 

-  6 

o 

+47 

o 

0 

-  33 

o 

+239 

+  32 

o 

—237 

o 

2V—  3^ 

1.  1395 

I.I9O 

—  IO 

+  4 

+20 

+  I 

—  IO 

-48 

—20 

+  IOO 

+  47 

—  ii 

—  99 

—21 

2V—  2^ 

1.0500 

1-307 

+10 

o 

-69 

0 

o 

+  75 

—  3 

—342 

—  74 

o 

+341 

—  3 

3v—  se" 

I.I770 

I.I52 

-  4 

+  S 

+  4 

+  4 

—21 

-  28 

-16 

+  21 

+  27 

—  22 

—  20 

-17 

3V-4S" 

1.0828 

I.26I 

—  i 

o 

+18 

+  5 

—   I 

-    5 

—19 

+  89 

+    5 

—    I 

-89 

-19 

3V—  3^ 

1.  008 

1.392 

+  10 

o 

-  7 

o 

+    I 

+  57 

—  i 

—  35 

-  55 

+    I 

+  35 

—  i 

2M—  2^ 

1.2679 

1.0765 

—23 

o 

+  3 

o 

+  3 

-116 

o 

+  17 

+  121 

O 

-  16 

o 

2M—  g' 

1.1581 

I.I708 

+  9 

—  6 

—  9 

—  7 

+30 

+   42 

+35 

-46 

-  42 

+29 

+  44 

+34 

4M—  3g" 

1.2602 

I.082I 

+   2 

—   2 

o 

o 

+20 

+  19 

+  4 

3 

-    19 

+19 

+    3 

+  4 

1-2S" 

1.3976 

0.9979 

+  9 

-  6 

—  9 

o 

-  5 

-  38 

+  i 

—  i 

+   36 

5 

0 

+  i 

J-/ 

1.2653 

1.0784 

-8  1 

0 

+  10 

o 

-13 

—416 

o 

+  59 

+415 

—13 

-  59 

o 

J 

I-I559 

I.I730 

—  i 

+16 

+  i 

+16 

-74 

—      2 

-84 

+  14 

+      I 

-74 

—  13 

—84 

2J—  3g~ 

I-S456 

0.9340 

+  3 

0 

o 

o 

—   2 

+  19 

+  i 

0 

—  19 

—   2 

o 

+  i 

2J—2g' 

1.3854 

1.0042 

+44 

o 

+  8 

0 

+  3 

+215 

o 

+  45 

—211 

+  3 

—  45 

o 

zj—g" 

1.2553 

1.0858 

+15 

5 

—  4 

—  4 

+34 

+  81 

+  8 

—  13 

-  81 

+34 

+  13 

+  8 

3J—  3g" 

1-5307 

0.9395 

+17 

0 

—  3 

o 

—  3 

+  73 

o 

-  19 

—  72 

-  3 

+  18 

0 

3J—  2g" 

1-3734 

i.  0106 

0 

+  10 

o 

—  I 

—42 

3 

+  7 

o 

+    3 

—42 

o 

+  7 

M-i* 

i.  2455 

1.0933 

o 

0 

o 

o 

—  IO 

+      2 

—  2 

o 

—      2 

—10 

o 

—  2 

ACTION  OF  THE  PLANETS  ON  THE  MOON. 


LUNAR  ARGUMENT  zD  =  ig  +  in  —  2g' . 


Planetary 
Argument. 

V 

V> 

I. 

(i 

I.' 

? 

2e, 

2ec 

»*.' 

2*.' 

2<r7r( 

2OTC 

2C7T/ 

2enc' 

v-S 

+•527 

+•554 

+   21 

0 

—  122 

o 

o 

o 

O 

—I 

+  2 

0 

—10 

o 

2V—  2? 

+.514 

+-569 

—  43 

o 

+  172 

0 

0 

—  T 

O 

+2 

—  4 

0 

+14 

o 

2M—  2^ 

+.562 

+•521 

+  59 

+   2 

—    II 

0 

o 

+i 

o 

—I 

+  5 

0 

—  i 

o 

J-e" 

+.56I 

+•521 

+212 

-  6 

-36 

0 

o 

+2 

0 

—I 

+  17 

o 

—  3 

o 

J 

+•539 

+•542 

+      I 

—39 

—    7 

—45 

+1 

O 

+1 

0    !   +   I 

-4 

—  I 

-4 

21—2? 

+.584 

+-503 

—  106 

+   2 

—  23 

o 

0 

—2 

o 

o 

—  9 

o 

3 

o 

LUNAR  ARGUMENT 


2n  —  2g'  . 


Planetary 
Argument. 

V 

v' 

I 

I 

i: 

V 

26, 

2Cc 

*'.' 

2eJ 

2ent 

2en, 

Mr/ 

2«r/ 

v-f 

+.346 

+.358 

+1 

O 

-4 

o 

O 

—  3 

0 

+22 

—  4 

o 

+23 

o 

2V—  2^ 

+•341 

+.364 

—I 

o 

+6 

o 

0 

+  8 

o 

—31 

+  8 

o 

—31 

o 

2tt—2g' 

+.361 

+•343 

+2 

0 

—i 

o 

o 

—ii 

o 

+   2 

—ii 

o 

+  2 

0 

J-f 

+.36I 

+•344 

+7 

o 

—  i 

o 

—I 

-39 

o 

+  7 

-38 

+1 

+  7 

o 

2J—2S" 

+•370 

+.336 

—3 

o 

—  i 

o 

0 

+18 

o 

+  5 

+18 

o 

+  5 

o 

LUNAR  ARGUMENT  i\  —  2D. 


r. 

rc 

r/ 

r/ 

r. 

rc 

r/ 

r/ 

v-g" 

0 

+13 

0 

-  5 

4X—3S' 

_! 

T 

—I 

+  I 

2V—  Sg" 

+1 

+  7 

+1 

—  4 

4M—  2g" 

+  1 

o 

+1 

+14 

2V—  2g" 

3V—  4^ 
3V—  Sf 
4V  —  4g' 

o 

+1 
+1 

o 

—14 
+  4 

+  2 

o 

o 
o 

+2 

o 

+17 
—  i 

—  2 

—24 

1-21? 
1-8? 

—  I 
O 

+6 

o 

+11 
+  I 

o 

+1 

+6 

—  I 

—19 

0 

2  J  —  3£/ 

o 

0 

o 

o 

M—  g' 

o 

+  I 

o 

—   I 

2J—  2^ 

0 

+25 

0 

+  7 

2M—2g' 

o 

+  3 

o 

—  5 

2j—  e" 

—I 

—  2 

—2 

+  4 

2V.—  g 

o 

—  3 

o 

-  4 

3J—  3^ 

o 

-  6 

0 

+  2 

3M—  3^ 

o 

o 

o 

0 

3J—  2^ 

-3 

o 

+1 

o 

3M—  2g" 

—I 

—  i 

—I 

+  1 

3J-S- 

o 

o 

+1 

o 

§  76.  Reduction  to  inequalities  of  true  longitude  and  collection  of  results. 

To  complete  the  work  it  is  necessary  to  transform  the  elemental  inequalities  into 
inequalities  of  the  coordinates.  As  already  remarked,  the  parallax  appears  to 
contain  no  sensible  terms  arising  from  the  action  of  the  planets;  only  inequalities 
of  longitude  and  latitude  are  therefore  considered.  In  the  case  of  terms  of  very 
long  period  the  transformation  to  true  longitude  is  unnecessary,  because  these  terms 
can  best  be  used  and  compared  as  elemental  inequalities.  A  precise  classification 
can  not,  however,  be  made  between  the  terms  which  are  to  be  transformed  and  those 
which  are  not.  What  has  actually  been  done  is  to  retain  as  elemental  inequalities 
those  depending  on  the  longitude  of  the  Moon's  node,  because  though  they  may 


INEQUALITIES  OF  ELEMENTS.  155 

ultimately  be  transformed  for  use  into  the  inequalities  of  the  coordinates,  they  are 
to  be  combined  with  terms  arising  from  the  compression  of  the  Earth  having  the 
same  argument.  The  two  Venus-terms  of  very  long  period  have  not  been  trans- 
formed because,  as  already  remarked,  they  can  be  most  conveniently  applied  to  the 
elements.  To  transform  the  other  terms  put  Sv,  the  perturbations  in  longitude  in 
orbit.  Then 

v  =  I  +  2e  sin  g  +  %#  sin  2g 
Sv  =  8/  -f  zSe  sin  g-  +  $e8e  sin  2g  +  2eSg  cos  g  -f  \  e2Sg  cos  2g 

Substituting 
81=  lc  cos  G  +  /,  sin  G          STT  =  TT.  cos  G  +  irt  sin  G          Be  =  ec  cos  G  +  et  sin  G 

SI—$Tr=8g=g-tcos  G+G,sin  G 
we  shall  have 

Sv  =  S/  +  2<?f  sin  G  sin  g  +  2ee  cos  G  sin  g 

+  2egc  cos  G  cos  g  +  2egt  sin  G  cos  g 
+  \eea  sin  G  sin  2g  +  ^cec  cos  G  sin  2g 
+  \  e2gc  cos  G  cos  2g  +  f  £2g",  sin  G  cos  2^ 
=  g/  _  (e,  -  egc)  cos  (6!  +  g)  +  (ec  +  eg)  sin  (G  +  g) 
+  («.  +  eg.)  cos  (G-g)-  (ec  -  eg,)  sin  (G  -  g) 
-  \e(e,  -  eg)  cos  (G  +  2g)  +  \e(ec  +  eg,)  sin  (G  +  2g) 
cos    ^  -  2      -       «.  -  eg'.    sin    £  -  2^-) 


In  nearly  or  quite  all  cases  we  may  drop  terms  of  the  second  order  in  e  and  use 

Sv  =  lc  cos  G  +  l.sinG  +  [_e(lc  -  TT,)  -  «.]  cos  (G  +  g)  +  [>(/.  -  TT,)  +  ee]  sin  (G  +  *) 

T«)  +  «.]  cos  (^  -  ^)  +  [«(/.  -  TT.)  -  e,-]  sin(G-  g) 


The  subsequent  processes  are  so  simple  and  familiar  as  to  scarcely  need  statement. 
All  terms  of  8v  depending  on  the  same  argument  are  combined  into  two,  one 
depending  on  the  sine,  the  other  on  the  cosine  of  the  argument.  Their  values  are 
shown  for  each  argument  in  the  following  table.  The  two  terms  are  then  combined 
into  a  monomial  satisfying  the  equation 

vt  sin  G  +  vc  cos  G  =  Sv  sin  (G  +  A) 

Terms  of  which  the  coefficient  St>  was  less  than  o".oo3,  have  generally,  but  not 
always,  been  dropped.  It  will  be  seen  that  even  exceeding  this  limit  there  are 
more  than  150  periodic  inequalities.  These  are  so  arranged  that  any  one  argument 
can,  it  is  hoped,  readily  be  found  on  a  system  which  will  be  evident  by  a  little 
examination. 

The  constituents  of  the  arguments,  including  ir,  are  all  measured  from  the 
Earth's  perihelion  (7r  =  99°.5).  The  secular  variations  of  the  coefficients  of  the 
periodic  terms  are  omitted,  because  they  can  better  be  derived  by  varying  the 
eccentricity  of  the  Earth's  orbit  in  the  expressions  for  the  inequalities  due  to  the 
Sun's  action. 


156 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


PERIODIC  INEQUALITIES  OF  THE  TRUE  LONGITUDE. 
ACTION  OF  VENUS. 


Argument. 

v. 

«i 

9f 

A 

Argument. 

z>c 

v, 

dv 

A 

V—  2g 

II 
—.001 

+.055 

n 

.055 

359°o 

g+2TT  —  3V+g" 

.000 

—.014 

.014 

0 

180.0 

V-g- 

+.OOI 

-.882 

.882 

179.9 

g+2ir—3V+2gr 

+  .012 

+.051 

•053 

13.2 

V 

—.000 

—  .014 

.014 

184.0 

g+2TT—3V+3g' 

+.014 

+.016 

.021 

41.2 

2V—  4^ 

—.004 

+.003 

.005 

306.8 

g+2TT—2V 

+  .OOI 

-.174 

.174 

180.0 

2V—  Sg* 

+.096 

—•340 

•354 

164.3 

g+2ir—2V+g' 

+  .OII 

+.070 

.071 

9.0 

2V—  2^ 

+.005 

+401 

401 

0-7 

g+2ir—  v  —g" 

.000 

+.146 

.146 

O.O 

3V  -Sg" 

+.060 

—.056 

.082 

133-0 

g+2ir+  V  —Sg" 

.000 

-.025 

.025 

180.0 

3V-4/ 

+.040 

—.191 

.197 

168.3 

g+2ir+2V—Sg' 

+  .011 

—  .040 

.041 

164.5 

3V—  3^ 

+.001 

—.037 

•037 

178.4 

g+2ir+2V—4g' 

.000 

+.142 

.142 

O.O 

Sv-Sg' 

+.018 

-.015 

.023 

129.8 

g+2ir+3V-7g' 

+.017 

—.019 

.026 

138.2 

g+2w+3v-sg' 

+.OO2 

-.646 

.646 

179.8 

g-Sv+Sg" 

+.004 

+.003 

.005 

53-2 

g+2Tr+Sv-8g' 

+.005 

—.024 

.025 

168.3 

e-sv+sg' 

.000 

+.008 

.008 

O.O 

g+2ir+6v—  log' 

—.051 

+.066 

.083 

322.3 

g—3V+4g' 

+.008 

+.040 

.041 

II-3 

g—SV+Sg* 

+.015 

+.018 

.023 

39-8 

g—2V+2g" 

+.OOI 

-.093 

•093 

180.0 

2g+2ir—3v+g' 

.000 

—•035 

•035 

180.0 

g-2V+3g' 

+.018 

+.080 

.082 

12.6 

2g+2lT—3V+2g' 

+.019 

+.090 

.092 

11.9 

2g+2ir—3V+3g' 

+.016 

+.020 

.026 

38-7 

g—v 

.000 

—.004 

.004 

180.0 

2g-\-21T—2V 

+.003 

-.142 

•  143 

178.8 

e—V+f 

.000 

+.166 

.166 

O.O 

2g+2ir—2V+g' 

+.O2O 

+.IOO 

.102 

1  1-3 

g-  V  +2g" 

.000 

—.003 

.003 

180.0 

2g+2lT—  V  —  g" 

.000 

+096 

.096 

0.0 

g+  V  —2g- 

.000 

+.003 

.003 

O.O 

2g+2ir+  v  -tf 

.000 

—.008 

.008 

180.0 

g+V  —g' 

.000 

—.149 

•149 

180.0 

2g+2lT+2V—5g' 

+.OIO 

—.048 

•049 

168.3 

g+v 

.000 

-.004 

.004 

180.0 

2g+21T+2V-4g' 

.000 

+.024 

.024 

O.O 

2g+2ir+3V—7gr 

+.O22 

—.027 

•035 

140.8 

g+2V—3g' 

+.018 

—.078 

.080 

167.0 

2g+2ir+3V—6g' 

+.OOI 

—.005 

.005 

168.7 

g-\-2V—2g" 

+.001 

+.071 

.071 

0.6 

2g+2ir+3V—5g' 

—.001 

+.056 

.056 

359-0 

e+SV-Sg" 

+.015 

—.016 

.022 

136.9 

«+3V-4«/ 

+.008 

—.036 

•037 

167-5 

g+3V-3g" 

.000 

-.006 

.006 

180.0 

3g+2ir—2V 

.000 

+.009 

.009 

0.0 

g+SV-Sg- 

+.004 

—.003 

.005 

126.8 

3g+2ir-  V  -g- 

.000 

-.006 

joo5 

180.0 

27T—  2V 

.000 

—.009 

.009 

180.0 

21T—  2V+g' 

+  .001 

+.003 

.003 

184 

4g+2TT—2V 

.000 

+.003 

.003 

O.O 

27T—  V  —ff 

.000 

+.007 

.007 

O.O 

4g+2TT—  V  —  g" 

.000 

—  .002 

.002 

180.0 

27T+2V  —  4g" 

.000 

+.008 

.008 

O.O 

27T+3V—  $g" 

.000 

+.072 

.072 

O.O 

2ir-  g  +3V-5/ 

.000 

+.003 

.003 

O.O 

27T+6V  —  lOg' 

-.037 

+.050 

.062 

323-5 

2TT—   g  +6V  —  IO/ 

—.002 

+.002 

.003 

315-0 

INEQUALITIES  OF  THE  TRUE   LONGITUDE. 


'57 


ACTION  OF  MARS. 


Argument. 

vc 

v. 

8v 

A 

Argument. 

Vc 

f. 

8v 

A 

H 

n 

0 

II 

II 

II 

o 

M-g- 

+.OOI 

—.Oil 

.Oil 

174-3 

2ir—  2M—  g' 

.000 

+.OO2 

.002 

O.O 

M 

—.OOI 

—.030 

.030 

l8l.9 

2M  —  2g" 

.OOO 

—.224 

.224 

180.0 

27T—  2M—  4^ 

.000 

—.004 

.004 

180.0 

2M—g' 

—-IQ3 

+.317 

•372 

328.7 

3M—3S' 

.OOO 

+.014 

.014 

0.0 

S+21T—  4M 

+  .011 

—.OOI 

.Oil 

95-2 

3M-2g' 

—.031 

+.029 

.042 

313.1 

g+2ir—  4M+/ 

—.004 

—.003 

.005 

233.2 

4M—3g' 

—.035 

+.032 

.047 

312.4 

£+2?r—  3M 

—.004 

—.004 

.006 

225.0 

4M—  2^ 

+.095 

+  .OO2 

•095 

88.8 

g+27T-3M+g/ 

.000 

+.006 

.006 

o.o 

SM—  3^ 

—  .020 

+.023 

.030 

319.0 

2+27T—  2M+^ 

—.026 

—.034 

•043 

217.4 

£+27T—  2M 

.000 

+.017 

.017 

O.O 

f—  4M  +  22' 

+.017 

—.002 

.017 

96.8 

g+2lT—  M  —  g" 

.000 

+.004 

.004 

o.o 

S-4M+3/ 

—.007 

—.007 

.010 

225.0 

g+2TT+  M  —3^ 

.000 

—.005 

.005 

180.0 

2—  3ii+2g- 

-.006 

-.006 

.009 

225.0 

g+21T+2M—  4g' 

.000 

-.066 

.066 

180.0 

g—3M+3g' 

.000 

—.003 

.003 

180.0 

g+2ir+2U—3g' 

—.023 

+•034 

.041 

325.9 

g—2tt+g- 

—.035 

—  .O42 

•055 

2194 

g+2ir+3U—5gr 

.000 

+.003 

.003 

0.0 

g—2U+2g- 

.000 

+•043 

•043 

o.o 

g+2ir+3U—4g' 

-.006 

+.006 

.009 

315.0 

g—M 

.000 

+.003 

.003 

0.0 

g+2ir+4U—Sg' 

-.009 

+.009 

.013 

315.0 

g-M+g~ 

.000 

+.004 

JOO4 

0.0 

g+2ir+4U—4g' 

+.009 

+.OOI 

.009 

83.7 

g+  M  —  ^ 

.000 

—.004 

.004 

180.0 

g+M 

.000 

—.003 

.003 

180.0 

2g+2ir—4M+g' 

—.004 

—  .003 

.005 

233.2 

£+2M—  2g~ 

.000 

—.048 

.048 

180.0 

2g+2TT—2TA—g' 

—•035 

—.046 

.058 

217.3 

g+2U-g' 

—.036 

+.042 

•055 

3194 

2g+21T—2M. 

.000 

+.006 

.006 

0.0 

£+3M—  2^ 

—.006 

+.006 

.009 

3i5.o 

2g+21T+2ll—4g' 

.000 

—.050 

.050 

180.0 

2+3M—  3g' 

.000 

+.003 

.003 

o.o 

2g+2lT+2M—3g' 

—.030 

+.042 

.052 

324.5 

g+4U—  2g" 

+.017 

+.002 

.017 

83.2 

2g+2ir+4V—5g' 

—.020 

+.019 

.028 

313.5 

g+w—ss" 

—  .007 

+  .007 

.010 

315.0 

3g+2TT+2M—4g' 

.000 

+.004 

.004 

o.o 

'58 


ACTION  OF  THE  PLANETS  ON  THE  MOON. 


ACTION  OF  JUPITER. 


Argument. 

». 

*», 

8v 

A 

Argument. 

*. 

v, 

9v 

A 

'j-2g- 

// 

+.004 

—  !ois 

.016 

165.1 

g+2TT-3J 

—445 

tl 

—0.015 

0445 

26&JO 

j-e' 

+.015 

—.741 

•741 

178.8 

g+2TT-3]+g' 

.000 

+0.018 

0.018 

O.O 

J 

+.070 

+.169 

.183 

22.5 

g+2ir—2]—g' 

—.010 

—  0.016 

0.019 

2I2.O 

2J-3S' 

.000 

+.006 

.006 

O.O 

S+21T—2J 

—.005 

—1.140 

1.140 

180.2 

2J  —  2? 

—.001 

+.242 

.242 

359-8 

21—  g? 

—  -059 

+.183 

•193 

342.2 

g+2TT—  ]  —2g- 

+.062 

+0.008 

0.062 

82.7 

3J-3^ 

+.001 

+.000 

.009 

6.4 

S+2TT—  J   —g" 

.000 

+0.064 

0.064 

O.O 

3J—  2^ 

+•042 

—  002 

.042 

177.2 

g+2lt—  J 

+.OIO 

+0.015 

0.018 

33-7 

3J  -e" 

-.024 

+.005 

.024 

281.8 

g+2TT+  J  —4g" 

+  .001 

—0.018 

0.018 

176.8 

g+2TT+  J   —  3g" 

+.004 

—0.230 

0.230 

179.0 

g+2ir+  j  —2g" 

+.060 

—0.003 

0.060 

92.9 

e—  3J+gr 

+.OO2 

—.002 

.003 

135-0 

g+2ir+2j-4g' 

+.OOI 

+0.098 

0.098 

0.7 

g—3J+2g/ 

+.OIO 

.000 

.010 

90.0 

g+2V+2J—3g' 

—.018 

+0.045 

0.048 

338.2 

?-3J+3^ 

.OOO 

—.004 

.004 

180.0 

g+2TT+3J-Sg' 

+.001 

+0.030 

0.030 

1.9 

ff-2J+^ 

+.005 

—.032 

.032 

179.1 

g+2ir+3J—4g- 

+.021 

—  0.001 

O.O2I 

92.8 

£—  21+2^ 

.000 

—.045 

•045 

1  80.0 

g+2ir+3J—3g' 

+.004 

+O.OOI 

0.004 

75-9 

t-J+i' 

+.004 

+.140 

.140 

1.6 

e—  J  +2g' 

+.001 

+.006 

.006 

9-6 

2g+2ir-3J 

—  .007 

o.ooo 

0.007 

270.0 

*—  j 

+.036 

+.OII 

.038 

73-0 

2g+2ir-3J+g' 

.000 

—  0.018 

0.018 

180.0 

*+  J  —2? 

+.OOI 

—.006 

.006 

1704 

2g+2TT—2J—g' 

-.008 

—0.013 

0.015 

21  1.6 

g+j-g' 

+.004 

-.163 

.163 

178.6 

2g+2lT—2J 

.000 

+0.018 

0.018 

0.0 

e+'j 

+.036 

—  JOll 

.038 

107.0 

2g+2lT—  J  —2g" 

+.040 

+0.006 

0.040 

81.5 

g+2J—2g' 

.000 

+.064 

.064 

O.O 

2g+2ir—  J  —g" 

.000 

+0.016 

0.016 

O.O 

£+21—  g1 

+.005 

+.036 

.036 

7-9 

2£+27T+  J  —  4£/ 

+.005 

—0.038 

0.038 

172.5 

g+SJ—Sg* 

JOOO 

+.003 

.003 

O.O 

2g+2ir+  J  —Sg" 

+.006 

-0.168 

0.168 

177.9 

g+3J—2g' 

+.OIO 

.000 

.010 

90.0 

23+21T+  J  —2g- 

+.036 

O.OOO 

0.036 

90.0 

g+SJ-g' 

+.OO2 

+.002 

.003 

45-0 

2g+2ir+2]—Sg" 

+.OO2 

+0.019 

0.019 

6.0 

2g+2TT+2]  —4g" 

—.001 

+0.092 

0.092 

359.3 

2g+2ir+2J—3g' 

-.035 

+0.082 

0.089 

336.9 

21T—3J 

-.258 

-.008 

.258 

268.2 

2g+2TT+3J—5g' 

+.003 

+0.074 

0.074 

2-3 

2TT—2J 

.000 

+.256 

.256 

O.O 

2g+2TT+3J—4g' 

+.042 

—0.003 

0.042 

94.0 

2ir—  J  —2g" 

+.004 

.000 

.004 

90.0 

22+27T+3J—  3g" 

+.OIO 

+O.OO2 

0.010 

78.5 

2*+  J  —  3^ 

.000 

—  .on 

.on 

180.0 

35+21T+  J  —Sg" 

.000 

+O.OII 

O.OII 

O.O 

27T+  J  —2^ 

+.004 

.000 

.004 

90.0 

3g+2ir+2J—4g' 

.000 

—0.006 

0.006 

180.0 

2TT+2]—4g' 

.000 

+.004 

.004 

O.O 

27T—  g—  2J 

.000 

+O.OIO 

O.OIO 

0.0 

21T+2J—  3^ 

—.001 

+.003 

.003 

341.6 

2ir—  g—3J 

-.015 

—  O.OOI 

0.015 

183.8 

ACTION  OF  SATURN. 


Argument. 

v< 

»• 

to 

A 

Argument. 

V, 

», 

8v 

A 

II 

// 

II 

o 

II 

It 

a 

0 

S 

+.016 

+.048 

x>5i 

184 

£+S 

+.001 

+0.004 

0.004 

14.1 

s-gf 

XXX) 

—.040 

.040 

180.0 

g+S-g- 

.000 

—  0.008 

0.008 

180.0 

as—g' 

—.001 

+.011 

.on 

354-8 

25—2? 

.000 

+.008 

.008 

0.0 

g-s 

+.OOI 

—0.004 

0.004 

165.9 

4T-S+/ 

.000 

+0.008 

0.008 

O.O 

INEQUALITIES  OF  LONG  "  "PERIOD.  159 

§  77.  Inequalities  of  the  elements  -which  have  not  been  reduced  to  inequali- 
ties of  the  longitude, 

Mean  longitude. 


sin(8v—  13^'  +  86°.  4)  +  o".O3O  sin0  —  o".273  cos  0 


Longitude'  of  Perigee. 

TT  =  TTO  -f  vj  +  253".  22  T—  38".49r2  -  o".oi3  T3  +  o".tf  sin  (i8v  -  i6g'—g  +  228°.5) 
—  o".67  sin  (8v  —  13^'+  86°.4)  —  o".io  sin  0  +  o".8o  cos  0 


Longitude  of  Node. 

0  =,  00  +  #,/  -  i37".8s  T+  f'.62  Tz  +  o".oo262rs  +  2".55  sin  0  -  i7"-33  cos  0  (1800) 

-f  2".  31  sin  0  —  17".  36  cos  0  (1900) 

Sin  y2  Inclination. 
By  =  —  o".ii5  cos  0  —  o".f]6c)  sin  0  (1800) 

—  o".iO4  cos  0  —  o".77o  sin  0  (1900) 

Hence: 

Inclination. 

87=  —  o".230  cos  0  —  i".S39  sin  0  (1800) 

—  o".2o8  cos  0  —  i"-54i  sin  0  (1900) 

It  may  be  found  advisable,  in  the  construction  of  new  lunar  tables,  to  include 
also  the  term 

S/  =  o".2S6  sin  (27r  —  2/) 

in  the  mean  longitude.  The  effect  of  including  this  term  in  the  preceding  trans- 
formations is  that  the  Jovian  evection,  and  the  coefficient  of  the  term  of  argument 
27r  —  ^J  —  g,  have  each  received  the  increment  -|-o".oi4.  Hence,  if  the  term 
were  included  in  the  mean  longitude,  the  coefficient  of  the  Jovian  evection  would 
be  —  i".  154,  and  of  the  other  term  named  —  o".oo4. 

§78.  Remarks  on  the  Possibility  of  Unknown  Terms  of  Long  Period.  In  his 
Researches  on  the  Motion  of  the  Moon,  published  in  1878,*  the  author  found  that 
the  representation  of  the  Moon's  mean  longitude  during  the  period  from  1650  to 
1875  showed  a  discrepancy  between  existing  theory  and  observation  which  might 
be  represented  by  a  term  having  a  period  of  two  or  three  centuries,  and  a  coefficient 
of  about  15".  This  coefficient  may  be  somewhat  reduced  by  the  introduction  of 
the  improved  values  of  the  terms  of  short  period  now  available,  but  it  does  not 
seem  likely  that  the  deviation  can  be  brought  below  10".  One  hypothesis  on 
which  the  discrepancy  might  be  explained  is  that  of  minute  fluctuations  in  the 

*  Washington  Obseri'ations  for  1875,  App.  II,  p.  268.  See  also  Monthly  Notices,  Royal  Astronomical  Society, 
vol.  i.xiri.  March,  1903,  p.  316. 


160  ACTION   OF  THE   PLANETS  ON  THE  MOON. 

Earth's  diurnal  rotation,  which  might  be  produced  by  the  motion  of  solids  and  fluids 
on  its  surface.  Observations  of  transits  of  Mercury  leave  scarcely  more  than  a 
possibility  of  changes  in  the  measure  of  time  having  the  magnitude  required  to 
explain  the  deviation.  The  observed  phenomena,  therefore,  point  very  strongly 
to  the  inference  that  there  must  be  some  term  of  long  period  still  undiscovered 
in  the  actual  mean  motion  of  the  Moon.  The  preceding  researches  seem  to 
remove  the  possibility  that  there  can  be  any  undiscovered  term  in  the  action  of  the 
planets.  It  is  true  that  there  are  two  possible  classes  of  inequality  which  are  not 
considered  in  the  present  work.  One  of  these  has  the  solar  parallax  as  a  factor, 
and  may  arise  from  two  sources;  one  the  development  of  the  potential  to  terms  of 
higher  order  than  the  principal  ones;  the  other  to  the  parallactic  terms  in  the 
Moon's  coordinates.  The  author  had  intended  to  carry  the  development  of  R  and 
flp  one  step  further,  so  as  to  include  these  terms.  But,  on  examining  the  periods 
of  the  inequalities  that  might  thus  arise,  none  were  found  that  could  lead  to  any 
important  term. 

Yet  another  class  of  terms  comprises  those  of  the  second  order  arising  from  the 
action  of  the  planets  being  modified  by  their  mutual  perturbations.  An  examina- 
tion which  I  believe  to  be  exhaustive  was  therefore  made  for  terms  of  long  period 
of  this  class.  None  have  been  found,  and  the  writer  believes  that  none  can  exist 
more  important  than  one  of  o".oi8  computed  by  Radau.  This  term  has  the  argu- 
ment $S  —  ij  of  the  great  inequality  between  Jupiter  and  Saturn.  In  this  connec- 
tion it  may  be  again  remarked  that,  in  determining  the  action  of  Venus  in  the 
present  work,  the  mutual  perturbations  of  Venus  and  the  Earth  have  been  taken 
account  of.  But  no  change  is  thus  produced  except  in  the  Hansenian  term  of  long 
period. 


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